1. (2015/Myanmar/q3 )
The four angles of a quadrilateral are in A.P. Given that the value of the largest angle is three times the value of the smallest angle, find the values of all four angles. (3 marks)
2. (2015/Myanmar/q3or )
(OR) The second term of a G.P. is 64 and fifth term is 27 . Find the first 6 terms of the G.P. $(3$ marks)
3. (2015/Myanmar/q9b )
The fourth term of an A.P. is 1 and the sum of the first 8 terms is 24 . Find the sum of the first three terms of the progression. $(5 \mathrm{~m} \mathrm{rks})$,
4. (2015/Myanmar/q10a )
The sum of the first three terms of a G.P. is 27 and the sum of the fourth, fifth and sixth terms is $-1$. Find the common ratio and the sum to infinity of $\begin{aligned}&\text { the G.P. } & & (5 \text { marks })\end{aligned}$
5. (2015/FC/q3or )
(OR) Write down the next two terms of the sequence $\sqrt{2}, \sqrt{10}, 5 \sqrt{2}, 5 \sqrt{10}, \cdots$ and defermine the $n^{\text {th }}$ term of the sequence. $\quad$ (5 marks)
6. (2015/FC/q9b )
The sum of four consecutive numbers in an A.P. is 28 . The product of the second and third numbers exceeds that of the first and last numbers by 18 . Find the numbers. $\quad(5$ marks $)$
7. (2016/Myanmar/q3or )
(OR) A geometric progression is such that the sum of the first 3 terms is $0.973$ times the sum to infinity. Find the common ratio.
8. (2016/Myanmar/q9b )
The third term of an A.P. is 9 and the seventh term is $49 .$ Calulate the thirteenth term. Which term of the progression, if any, is $289 ?$
9. (2016/Myanmar/q10a )
The first and second terms of a G.P. are 10 and 11 respectively. Find the least number of terms such that their sum exceeds 8000 .
10. (2016/FC/q3 )
If $29, a-b, a+b, 95$ is an A.P., find the values of $a$ and $b$.
11. (2016/FC/q3or )
(OR) In an G.P, the ratio of the sum of the first three terms to the sum to infinity of the G.P. is $19: 27$. Find the common ratio.
12. (2016/FC/q9b )
Find the sum of all two-digit natural numbers which are not divisible by $3 .$
13. (2016/FC/q10a )
Find the sum of $(b+2)+\left(b^{2}+5\right)+\left(b^{3}+8\right)+\ldots$ to 18 terms in terms of $b$ where $b \neq 1$
14. (2017/Myanmar/q3 )
The $p$ th term of an A.P. is $q$ and the $q$ th term of this A.P, is $p$. Show that its $(p+q)$ th term is zero.
Q3 Solution
15. (2017/Myanmar/q3or )
(OR) If the first term of a GP. exceeds the second term by 2 and the sum of infinity is 50 . Find the first term and the common ratio.
Q3(OR) Solution
16. (2017/Myanmar/q9b )
If $b, x, y, c$ are consecutive terms of a G.P. and the A. $M$. between $b$ and $c$ is $a$, then prove that $x^{3}+y^{3}=2 a b c$
Q9(b) Solution
17. (2017/Myanmar/q10a )
The product of first three terms of a G.P is 1000 . If we add 6 to its second term, 7 to its third term and its first term is not changed, then three terms form an A.P. Find the first three terms of the G.P.
Q10(a) Solution
18. (2017/FC/q3 )
Given that $\sin ^{2} x, \cos ^{2} x$ and $5 \cos ^{2} x-3 \sin ^{2} x$ are in A.P., find the value of $\sin ^{2} x$. (3 marks)
19. (2017/FC/q3or )
(OR) If $\log _{2} 3, \log _{5} \mathrm{x}, \log _{3} 16$ is a G.P., then find the possible values of $\mathrm{x}$. (3 marks)
20. (2017/FC/q9b )
Let a and $b$ be two numbers, $x$ be the single arithmetic mean of $a$ and $b$. Show that the sum of $\mathrm{n}$ arithmetic means between a and $\mathrm{b}$ is $\mathrm{nx}$. $\quad$ (5 marks)
21. (2017/FC/q10a )
The three numbers $\mathrm{a}, \mathrm{b}, \mathrm{c}$ between 2 and 18 are such that their sum is 25 , the numbers $2, \mathrm{a}, \mathrm{b}$ are consecutive terms of an arithmetic progression, and the numbers $\mathrm{b}, \mathrm{c}, 18$ are consecutive terms of a geometric progression. Find the three numbers. ( 5 marks)
22. (2018/Myanmar/q3 )
In an A.P. whose first term is $-27$, the tenth term is equal to the sum of the first nine terms. Calculate the common difference.
Click for Solution
23. (2018/Myanmar/q3or )
(OR) The first term of a G.P. is $a$ and the common ratio is $r$. Given that $a=12 r$ and that the sum to infinity is 4 , find the third term.
Click for Solution
24. (2018/Myanmar/q9b )
An A.P. is such that the 5 th term is three times the 2 nd term. Given further that the sum of the $5 \mathrm{th}, 6$ th, 7 th and 8 th terms is 240 , calculate the value of the first term.
Click for Solution
25. (2018/Myanmar/q10a )
A G.P. of positive terms and an A.P. have the same first term. The sum of their first terms is 1 , the sum of their second terms is $\frac{1}{2}$ and the sum of their third terms is 2 . Calcualte the sum of their fourth terms.
Click for Solution
26. (2018/FC/q3 )
If $x, y, z$ is a G.P., show that $\log x, \log y, \log z$ is an A.P.
27. (2018/FC/q3or )
(OR) If $\log x, \log y, \log z$ is an A.P., show that $x, y, z$ is a G.P.
28. (2018/FC/q9b )
An A.P. contains 25 terms. The last three terms are $\frac{1}{x-4}, \frac{1}{x-1}$ and $\frac{1}{x}$. Calculate the value of $x$ and the sum of all the terms of the progression.
29. (2018/FC/q10a )
If $a, b, c$ be in A.P., and if $u, v$ be the A.M. and G.M. between $a$ and $b, x, y$ the A.M. and G.M. between $b$ and $c$, then prove that $u^{2}-v^{2}=x^{2}-y^{2}$.
30. (2019/Myanmar/q2b )
Find the sum of all even numbers between 69 and 149 . (3 marks) Click for Solution
31. (2019/Myanmar/q8b )
Find the sum of the first 12 terms of the A.P. 44,$40 ; 36, \ldots \ldots .$ Find also the sum of the terms between the $12^{\text {th }}$ term and the $26^{\text {th }}$ term of that A.P. $\quad$ (5 marks) Click for Solution
32. (2019/Myanmar/q9a )
9. The sum of the first n terms of a certain sequence is given by $\mathrm{S}_{\mathrm{n}}=2^{\mathrm{n}}-1$. Find the first: 3 terms of the sequence and express the $\mathrm{n}^{\text {th }}$ term in terms of n. $\quad$ (5 marks) Click for Solution
33. (2019/FC/q2b )
How many terms of the arithmetic progession $9,7,5, \ldots .$ add up to $24 ? \quad$ (3 marks) Click for Solution 2(b)
34. (2019/FC/q8b )
The sum of the first six terms of an A.P. is 96 . The sum of the first ten terms is one-third of the sum of the first twenty terms. Calculate the first term and the tenth term. ( 5 marks) Click for Solution 8(b)
35. (2019/FC/q9a )
9. The sum of the first n terms of a certain sequence is given by $\mathrm{S}_{\mathrm{n}}=\frac{1}{2}\left(3^{\mathrm{n}}-1\right)$. Find the first 3 terms of the sequence and express the $\mathrm{n}$ " term in terms of n. (5 marks ) Click for Solution 9(a)
36. (2015/FC/q10a )
A geometric progression has three terms $a ; b, c$ whose sum"is 42 . If 6 is added to each of the first two terms. and 3 to the third, a new G.P. results whose first term is the same as $b$. Find $a, b$ and $c . \quad(5$ marks $)$
Answer (2015-2019)
1. 45,75,105,135
2.or 256/3,64,48,36,27,81/4
3. $S_3=-21$
4. $r=-\frac 13, S=\frac{729}{28}$
5.or $25\sqrt 2,25\sqrt{10},u_n=(\sqrt 5)^{n-1}\sqrt 2$
6. $11\frac 12,8\frac 12,5\frac 12,2\frac 12$
7.[O R] $\quad r=0.3$
8. $u_{13}=109,n=31$
9. $47$
10. $a=62, b=11$
11.(OR) $r=\frac{2}{3}$
12. 3240
13. $495+\frac{b\left(1-b^{18}\right)}{1-b}$
14. Show
15.or $a=10, r=\frac{4}{5}$
16. Prove
17. $5,10,20$ or $20,10,5 \quad$
18. $\dfrac{3}{5}$
19. OR $x=25$ or $\dfrac{1}{25}$
20. Prove
21. 5,8,12
22. $d=8 \quad$
23. (OR) $u_{3}=\frac{3}{16}$
24. $a=5$
25. $\frac{19}{2}$
26. Show
27.(OR) Show
28. $x=-2,37.5$
29. Prove
30. 4360
31. $S_{12}=264,-364$.
32. 1,2,4, $u_n=2^{n-1}$
33. 4 or 6
34. $a=11,u_{10}=29$
35. $1,3,9,\ldots$ $u_n=3^{n-1}$
36. $a=6,b=12,c=24$
Group (2014)
1. Which term of the arithmetic progression $13,20,27, \ldots$ is 111 ? (3 marks)
2. An A.P. contains 25 terms. If the first term is 15 and the last term is 111 , find the middle term. (3 marks)
3. How many terms of the A.P. $3,5,7, \ldots$ gives a sum of $224 ?$ (3 marks)
4. If the $n^{\text {th }}$ term of an A.P. is $4 n-7$, find the sum of the first 40 terms. (3 marks)
5. The eighth term of an A.P. is 150 and the fifty-third term is $-30$. Determine the number of terms whose sum is zero. (3 marks)
6. The $6^{\text {th }}$ term of an A.P. is 21 and the sum of the first 17 terms is 0 . Find the $3^{r d}$ term of the A.P. (3 marks)
7. The $n^{\text {th }}$ term of an A.P. is $p$ and the $(n+1)$ th term is $q$. Find the first term and the fifth term in terms of $n, p$ and $q$. (3 marks)
8. If the ratio of the sums of the first $m$ terms and the first $n$ terms of an A.P. is $m^{2}: n^{2}$, then find the ratio of its $5^{\text {th }}$ term and $8^{\text {th }}$ term. (3 marks)
9. If $S_{1}, S_{2}, S_{3}$ are the sum of $n, 2 n, 3 n$ terms of an $A . P$, then show that $3\left(S_{2}-S_{1}\right)=S_{3}$ (3 marks)
10. The sum of four consecutive numbers in an $A . P$ is 38 . The product of the second and third numbers exceeds that of the first and last by $18 .$ Find the numbers. (5 marks)
11. In an A.P., the first term is $-5$ and the last term is 91 . If the sum of the whd series is 1075, find the number of terms and the commoon difference. (5 marks)
12. The number of terms in an $A . P$, is 40 and the last tenn is -54. Given that the sum of the first 15 terms added to the sum of the first 30 terms is zero, calculate the sum of the progression. (5 marks)
13. The common difference of an A.P. is 3. The sum of the first 12 terms is 174 and the sum of the whole series is 850 . Calculate the first term, the number of terms and the last term. (5 marks)
14. In an A.P., the sum of the first 6 terms is 93 and the sum of the next 3 ferms is 114. Find the tenth term. (5 marks)
15. In an A.P., the sum of the first 6 terms is 93 and the sum of the next 3 terms is 114. Find the $21^{\text {st }}$ term. (5 marks)
16. A certain A.P has 25 terms. The last three terms are $\frac{1}{x-4}, \frac{1}{x-1}$ and $\frac{1}{x}$. Calculate the value of $x$ and the sum of all the terms of the progression. (5 marks)
17. If the sum of first 6 terms of an $A . P$. is 42 and the first term is 2, find the common difference. If the sum of first $2 n$ terms of that $A . P$. exceeds the sum of frist $n$ terms by 154 , find the value of $n$. (5 marks)
18. Which term of the progression $19+18 \frac{1}{5}+17 \frac{2}{5}+\ldots$ is the first negative term? What is the smallest number of terms which must be taken for their sum to be negative? Calculate this sum exactly. (5 marks)
19. If $\frac{1}{a+b}, \frac{1}{2 b}, \frac{1}{c+b}$ are in A.P., then prove that $a, b, c$ are in G.P. (3 marks)
20. Find the sum of the first 12 terms of the series $2+3+2^{2}+3^{2}+2^{3}+3^{3}+\ldots$ (5 marks)
21. The $4^{\text {th }}$ term of a GP. is 1 and the $8^{\mathrm{ti}}$ is $\frac{1}{256}$. Find the $9^{\text {th }}$ term of the $G P$. (3 marks)
22. A GP. contains 11 terms. If the first term is 5 and the last term is 5120 , find the middle term. (3 marks)
23. If the $n^{\text {th. }}$ term of the G.P. $3, \sqrt{3}, 1, \ldots$ is $\frac{1}{243}$, then find the number of terms. (3 marks)
24. The lengths of a triangle are in a $G P$. and the longest side has a length of $36 \mathrm{~cm}$. Given that the perimeter is $76 \mathrm{~cm}$, find the length of the shortest side. (3 marks)
25. The ratio of the sum of the first 6 terms of $a G P$. to the sum of the first 3 term: is $35: 8$. Find the common ratio of the progression. (3 marks) (3 marks)
26. Determine whether the sum to infinity of the G.P $5,0.5,0.05, \ldots$ exists or not If it exists find it. (3 marks)
27. A geometric progression is defined by $u_{n}=\frac{1}{3^{n}}$. Find the sum to infinity. (3 marks)
28. Find the sum to infinity of the series $\left(\frac{1}{2}+\frac{1}{3}\right)+\left(\frac{1}{2^{2}}+\frac{1}{3^{2}}\right)+\left(\frac{1}{2^{3}}+\frac{1}{3^{3}}\right)+\ldots$. (3 marks)
29. If the first term of an infinite G.P. is 1 and each term is twice the sum of the succeeding terms, find the common raito. (3 marks)
30. The product of the first three terms of a $G P$. is 27 . The ratio of the sum of the first 6 terms of the $G P$, and the sum of the first 3 terms of the G.P. is $9: 1$. Find the tenth term of the $G P$. (5 marks)
31. The sum of the first five terms of a $G P$. is 4 and the sum of the terms from the fourth to the eight inclusive is $7 \frac{13}{16}$, Find the common ratio and the sixth term. (5 marks)
32. If the four numbers forming a $G P$. are such that the third term is greater than the first by 9 and the second term is greater than the fourth by 18 , then find the numbers. (5 marks)
33. The ratio of the sum of the first, second and third terms of a geometric progression to the sum of the third, fourth and fifth terms is $9: 16$. Find the tenth term of the progression if the sixth term is $15 \frac{3}{16}$. (5 marks)
34. If the sum of first 4 terms of a G.P. is 960 and the sum of first 8 terms is 1020 , find the first term and common ratio. Find also the sum to infinity if it exists. (5 marks)
35. The sum of the first two terms of a G.P. is 12 and the sum of the first four terms is $120 .$ Calculate the two possible values of the fifth term in the progression. (5 marks)
36. A GP. has first term 5 and last term 2560 . If the sum of all its terms is 5115, how many terms are these? (5 marks)
37. Find the smallest value of $n$ for which the sum to $n$ terms and the sum to infinity of a GP $1, \frac{1}{7}, \frac{1}{49}, \ldots$ differ by less than $\frac{1}{1000}$. (5 marks)
38. An infinite geometric series has a finite sum. Given that the first term is 18 and that the sum of the first 3 terms is 38 , calculate the common ratio and the sum to infinity where it exists. (5 marks)
39. A number consists of three digits in $G P$. If the sum of the right hand and left hand digits exceeds twice the middle digit by 1 and the sum of the left hand and middle digits is two third of the sum of the middle and right hand and digits, then find the number. (5 marks)
40. If $k$ is positive integer, show that the sum of the A.P. $3 k+2,3 k+5,3 k+8, \ldots$ $3 k+44$ is divisible by 15 . (5 marks)
41. If $T_{1}, T_{2}, T_{3}$ are the sums of $n$ terms of three series in $A . P$, the first term of each being 1 and the respective common difference being $1,2,3$, then show that $T_{1}+T_{3}=2 T_{2}$ (5 marks)
Answer (2014)
1. $u_{15}=111$
2. $63$
3. $n=14$
4. 3000
5. $n=90$
6. $u_3=42$
7. $a=(1-n) q+n p ; u_{5}=(5-n) q+(n-4) p$
8. $\frac{3}{5} \quad$
9. Show
10. $5,8,11,14 ; 14,11,8,5$
11. $n=25 ; d=4$
12. $-600$
13. $a=-2, n=25, u_{25}=70$
14. $u_{10}=48$
15. $u_{21}=103 \quad$
16. $x=-2, \frac{75}{2}$
17. $n=7, d=2$
18. $u_{25}, n=49, S_{49}=-\frac{49}{5}$
19. Prove
20. 1218
21. $u_{9}=\pm \frac{1}{1024}$
22. $\pm 160$
23. $n=13$
24. 16 cm
25. $r=\frac{3}{2} \quad$
26. The sum to infinity of G.P exists, $\mathrm{S}=\frac{50}{9}$
27. $\frac{1}{2}$
28. $\frac{3}{2}$
29. $r=\frac{1}{3}$
30. $u_{10}=3\times 2^8$
31. $r=\frac{5}{4}, u_{6}=\frac{3125}{2101}$
32. $3,-6,12,-24$
33. $u_{10}=48$
34. $a=512, a=1536, r=\pm \frac{1}{2}, S=1024$
35. $u_{5}=243$ (or) $-486$
36. $n=10$
37. $n=4$
38. $r=-\frac{5}{3}$ (or) $r=\frac{2}{3}, S=54$
39. 469
40. Show
41. Show
Group (2013)
1. The sixth term of an $A . P$ is 32 while the $10^{\text {th }}$ term is 48 . Find the $21^{\text {st }}$ term. (3 marks)
2. If $19, a-b, a+b, 85$ is an $A . P .$, find the values of $a$ and $b$. (3 marks)
3. If the $n^{\text {th }}$ term of an A.P. is given by $U_{n}=4 n-1$, find the sum of the first 10 terms. (3 marks)
4. The last term of an $A$.P. of 20 terms is 195 and the common difference is 5 . Find the sum of the progression. (3 marks)
5. Let $S_{5}$ be the sum of the first five terms of an A.P. and $S^{*}$ be the sum of the next five terms. If $S^{*}-S_{5}=75$, then find the common difference. (3 marks)
6. The $6^{\text {th }}$ term of an $A . P$. is 21 and the sum of the first 17 terms is 0 . Find the common difference of the $A . P$. (3 marks)
7. The first term of an $A . P$. is 2 and $n^{\text {th }}$ term is 20 . If the sum of the first $n$ terms of that $A$.P. is 110, find the value of $n$. (3 marks)
8. The first term of an $A \cdot P .$ is $-2$ and its $n^{\text {th }}$ term is $18 .$ If the sum of the first $n$. terms is 88 , find $n$ and the common difference. (3 marks)
9. For a certain $A . P, S_{n}=\frac{n}{2}(3 n-17)$. Find the first three terms of the corresponding sequence. (3 marks)
10. If the $n^{\text {th }}$ term of an A P. 2. $3 \frac{7}{8}, 5 \frac{3}{4}, \ldots$ is equal to the nth term of an A.P. $187,184 \frac{1}{4}, 181 \frac{1}{2}, \ldots$, find $n$. (5 marks)
11. The sum of the first 6 terms of an A.P. is 96 . The sum of the first ten terms is one-third of the sum of the first 20 terms. Calculate the first term and the tenth term. (5 marks)
12. The sum of the first six terms of an A.P. is 96 . The sum of the first ten terms is one-third of the sum of the first twenty terms. Calculate the first term and the tenth term. (5 marks)
13. The sum of the first $n$ terms of the A.P. 13, 16.5, 20,... is the same as the sum of the first $n$ terms of the A.P. $3,7,11, \ldots$ Find the value of $n .$ Find also the $n$-th term of the first $A . P$. (5 marks)
14. In a certain A.P., the first term is a and the common difference is $d$. If the third term is 5, find the sum of the first five terms of the A.P. Given that the tenth term is 33 , calculate the sum from the sixth term to the tenth term of that $A . P$. (5 marks)
15. The sum to $n$ terms of an $A . P$. is 21 . The common difference is 4 and sum to $2 n$ terms is 78 , find the first term. (5 marks)
16. The eleventh term of an A.P. is 32 and the sum from the fifth term to the eighth term of the A.P. exceeds the sum of the first four terms by 48 . Find the sum of the first 13 terms of the $A . P$. (5 marks)
17. The ninth term of an A.P. is 42 and the sum from the sixth term to the tenth term of the $A$.P. exceeds the sum of the first 5 terms of the $A . P$ by 100 . Find the sum of the first 15 terms of the A.P. (5 marks)
18. An A.P. contains thirteen terms. If the sum to first four terms is 32 and the sum of the last four terms is 176 , find the middle term of that $A . P$. (5 marks)
19. A polygon has 25 sides, the lengths of which starting from the smallest side are in $A . P$. If the perimeter of the polygon is $2100 \mathrm{~cm}$ and the length of the largest side 20 times that of the smallest, then find the length of the smallest side and the common difference of the $A . P$. (5 marks)
20. If the first, second and last terms of an $A$.P. are $a, b$ and $2 a$ respectively, then show that its sum is $\frac{3 a b}{2(b-a)}$. (3 marks)
21. An A.P is such that the $5^{\text {th }}$ term is three times the $2^{\text {nd }}$ term. Show that the sum of the first eight terms is four times the sum of the first four terms. (5 marks)
22. If $x, y, 2 x$ is an $A . P$. and $3,9, y$ is a $G P$., then find the values of $x$ and $y$. (3 marks)
23. There are four numbers of which the first three are in $G P$. and the last three are in $A . P$. whose common difference is 6 . If the first number and the last number are equal, then find the numbers. (5 marks)
24. The first three terms of an A.P are $x, y, z$. If these numbers $x, y, z$ are also the first, third and fourth terms of a $G P$. Show that $(2 y-z) z^{2}=y^{3}$. (5 marks)
25. The sum to the first $n$ terms of a series is $S_{n}=\frac{n}{2}(3 n+17)$, Calculate $u_{1}, u_{2}$ and $u_{3} .$ Hence show that it is an $A . P$. (3 marks)
26. Write down the next two terms of the sequence $\sqrt{2}, \sqrt{6}, 3 \sqrt{2}, 3 \sqrt{6}, \ldots$ and determine the $n^{\text {th }}$ term of this sequence. (3 marks)
27. In $\mathrm{a} G P$. the fourth term is 6 and seventh term is $-48$. Calculate the first term. (3 marks)
28. The sixth term of a geometric series of positive number is 10 and the sixteenth term is $0.1$. Find the eleventh term. (3 marks)
29. If $432, p-q, p+q, 2$ is a $G P$., find the values of $p$ and $q$. (3 marks)
30. Find ' $n$ ' if $1+2+2^{2}+\ldots+2^{n}=511$. (3 marks)
31. In a $G P$., the third term exceeds the first term by 16 . If the sum of the third term and the fourth term is 72 , find the common ratios. (3 marks)
32. Solve the equation $1+x+x^{2}+x^{3}+\ldots+x^{11}=x+3+\frac{x^{i 2}}{x-1}$. (5 marks)
33. The first term of a $G P$,, is a and the common ratio is $r$. Given that $a=12 r$ and the sum to infinity is 4 . Calculate the third term. (5 marks)
34. In an infinite $G P$., the sum to infinity is 20 and the sum to the first 3 terms is 22.5. Find the fifth term of the $G P$. (5 marks)
35. AGP. has first term 2 and common ratio $0.95$. Calculate the least value of $n$ for which $\mathrm{S}-\mathrm{S}_{n}<1$. (5 marks)
36. If $3^{3 x-1}, 9^{x}, 27^{3-x}$ are the first three terms of a $G P$., find the value of $x$. Find also the smallest positive integer $n$ such that the sum to infinity a $n$ terms of that $G P$. differ by less than $0.0005$. (5 marks)
37. The sum to infinity of a $G P$. is twice the first term. Find the common ratis. (3 marks)
38. The sum toinfinity of a $G P$. is 8 and the $2^{\text {nd }}$ term is 2 . Find the first, $1 . u$ and sixth terms of the $G P$. (3 marks)
39. In an infinite $G P$., the ratio of the sum to the first three terms and the sum to infinity is $37: 64$. Find the common ratio and the first term of the $G P$, if the third term is 81 . (3 marks)
40. An infinite geometric series has a finite sum of 256 . The sum of the first 3 terms is 224 . What is the value of the third term? (5 marks)
41. An infinite geometric series has a finite sum. Given that the first term is 18 and that the sum of the first 3 terms is 38 , calculate the value of the common ratio and the sum to infinity. (5 marks)
42. Given that $2 x-14, x-4$ and $\frac{1}{2} x$ are successive terms of a $G P .$ find the value of $x$. If $2 x-14$ is the third term of this $G P$. with infinite terms, find the sum to infinity. (5 marks)
43. A GP. has a first term of 16 and a sum to infinity of 24 . Given that each of the terms in the progression is squared to form a new $G P .$, find the sum to infinity of the new $G P$. (5 marks)
44. If $5^{\text {th }}, 8^{\text {th }}$ and $11^{\text {th }}$ terms of a $G P$. are $p, q$ and $s$ respectively, then show that $q^{2}=p s$. (3 marks)
Answer (2013)
1. 92
2. $a=52, b=11$
3. $210 \quad$
4. $2950 \quad$
5. $d=3$
6. $d=-7$
7. $n=10$
8. $ n=11, d=2 $
9. $-7 .-4,-1 $
10. $n=41$
11. $a=11, u_{10}=29$
12. $a=11, u_{10}=29$
13. $n=41 ;153$
14. $25 ; 125$
15. $a=3$
16. 260
17. 570
18. 26
19. $a=8, d=\frac{19}{3}$
20. Show
21. Show
22. $x=18, y=27$
23. 8,$-4$, 2 ,8
24. Show
25. 10,13,16
26. $9 \sqrt{2}, 9 \sqrt{6} \text { and } \sqrt{2 \times 3^{n-1}}$
27. $a=-\frac{3}{4}$
28. $u_{11}=1 \quad$
29. $p=42, q=-30 \quad$
30. $n=8 \quad$
31. $r=\frac{3}{2}$ (or) 3
32. $-1 \pm \sqrt{3}$
33. $\frac{3}{16}$
34. $\frac{15}{8}$
35. $72$
36. $13$
37. $r=\frac{1}{2} \quad$
38. $a=4, u_{3}=1, u_{6}=\frac{1}{8} \quad$
39. $r=\frac{3}{4}, a=144 \quad$
40. 32
41. $r=\frac{2}{3} ;54$
42. $x=16 ; 121.5 \quad
43. 288
44. Show
Group (2012)
| $\quad$ | $\,$ | |
|---|---|---|
| 1. | Find $u_{5}$ if $u_{n+2}=u_{n+1}+u_{n}-2^{n}$ with $u_{1}=3$ and $u_{2}=7.$ (3 marks) | |
| 2. | The sum of the first six terms of an $A.P.$ is 72 and the fifth term is 15. Find the first terms. (3 marks) | |
| 3. | The $n^{\text {th }}$ term of an $A. P.$ is $2 n+5.$ Find the sum of the first twenty terms. (3 marks) | |
| 4. | The sum of the first 8 terms of an $A.P.$ is 72 and the sum of the first 12 terms is 156. Find the first term and the common difference of the $A.P.$ (3 marks) | |
| 5. | The fifth term of $A.$P. is 24 and the sum of the first 8 terms is 172. Find the sum of the first 12 terms. (5 marks) | |
| 6. | In an A.P. whose first term is $-27$, the 10 -th term is equal to the sum of the first 9 terms. Calculate the common difference. Find also the sum from the 10 -th term to the 20 -th term. (5 marks) | |
| 7. | In an A.P. the first term is $-5$ and the last term is 91. If the sum of the whole series is 1075 , find the number of terms and the common difference. (5 marks) | |
| 8. | In an A.P., the $13^{\text {th }}$ term is 25 and the sum of the first 11 terms is 121. Find the first term and the common difference of that $A.$P. Show also that the sum of the first $n$ terms of that $A.$P. is $n^{2}.$ (5 marks) | |
| 9. | The sum to the first $n$ terms of an A.P. is 124 and the sum to $2 n$ terms is 440. If the common difference is 3, find the first term. (5 marks) | |
| 10. | An A.P. contains seven terms, the sum of the three terms in the middle is 39 and the sum of the last three terms is 57. Find the series. (5 marks) | |
| 11. | An A.P. contains 15 terms. If the sum of first five terms is 55 and the sum of last five terms is 255. Find the middle term. (5 marks) | |
| 12. | How many terms of the A.P. $5,7,9, \ldots$ give a sum of 320 ? (3 marks) | |
| 13. | The sum to the first 6 terms of an A.P. $2,5,8 \ldots$ is equal to the $n^{\text {th }}$ term of an $A.$P. $77,73,69, \ldots.$ Find $n.$ (3 marks) | |
| 14. | In an A.P., the sum of the first three terms is 21 and the sum of the first twelve terms is 192. Find the corresponding sequence. (3 marks) | |
| 15. | If the ratio of the sum of $m$ terms and $n$ terms of an A.P. is $m^{2}: n^{2}$, then show that the ratio of its $m^{t h}$ and $n^{t h}$ terms is $(2 m-1):(2 n-1).$ (3 marks) | |
| 16. | If the sum of the first $n$ terms of an A.P. is $2 n$ and the sum of the first $2 n$ terms is $n.$ Find the sum of the first $4 n$ terms. (5 marks) | |
| 17. | If the ratio of the sum of $n$ terms of two arithmetic progressions is $(3 n-13):(5 n+21)$, then find the ratio of $24^{\text {th }}$ terms of the two progressions. (5 marks) | |
| 18. | The fourth and sixth terms of an $A.P.$ are $x$ and $y$ respectively. Show that the $10^{\text {th }}$ term is $3 y-2 x.$ (3 marks) | |
| 19. | The first term of an $A.P.$ is 2 and its $n^{\text {th }}$ term is 20. If the sum of the first $n$ terms of that A.P: is 110, find the value of $n.$ (3 marks) | |
| 20. | The sum of the first $n$ terms of a certain $A. P.$ is $S_{n}=n^{2}+5 n$, find the $n^{\text {th }}$ term in terms of $n.$ (3 marks) | |
| 21. | The sum of four consecutive numbers in an A P.P. is 28. The product of the second and third numbers exceeds that of the first and last by 18. Find the numbers. (5 marks) | |
| 22. | The first 3 terms of an arithmetic progression are $4 p^{2}-10,8 p$ and $4 p+3$ respectively. Find two possible values of $p.$ If $p$ is positive and then the nth term of the progression is $-93$, find the value of $n.$ (5 marks) | |
| 23. | Show that $\sqrt{2}+\sqrt{6}+3 \sqrt{2}+3 \sqrt{6}+\ldots$ to 6 terms $=13(\sqrt{6}+\sqrt{2}).$ (5 marks) | |
| 24. | If $2 t+4, t+5$ and $t+1$ are three consecutive terms of a $G P.$ then find the values of $t.$ (3 marks) | |
| 25. | If $2 p+q, 6 p+q, 14 p+q$ are the first three terms of a G.P. with $p \neq 0$, find $q$ in terms of $p$ and the common ratio. (3 marks) | |
| 26. | If the sum of the first three terms of a GP. is 21 and the sum of the next three terms is 168 , then find the first term and the common ratio. (3 marks) | |
| 27. | The lengths of the sides of a triangle form $\mathrm{a} G P.$ If the largest side is $18 \mathrm{~cm}$ and the perimeter is $38 \mathrm{~cm}$, find the lenghts of the other two sides. (3 marks) | |
| 28. | The product of the first three terms of a $G P$ is 8 and the product of the second, third and fourth terms is 27. Find the fifth term of the $G P.$ (5 marks) | |
| 29. | The product of the first three terms of a $G P.$ is 1 and the product of the third, fourth and fifth terms is $11 \frac{25}{64}.$ Find the $8^{\text {th }}$ term of $G P.$ (5 marks) | |
| 30. | The third term of a $G P$ is 28 and the sum of the first three terms is 133. If the terms are all positive, calculate the value of common ratio and the sum to infinity. (5 marks) | |
| 31. | The second term of a $G P.$ is 2 and the sum infinity is $9.$ Find the sum of the first six terms of the $G P.$ (5 marks) | |
| 32. | The three numbers $a, b, c$ between 2 and 18 are such that their sum is 25, the numbers $2, a, b$ are consecutive terms of an $A. P.$, and the numbers $b, c, 18$ are consecutive terms of a $G P.$ Find the three numbers. (5 marks) | |
| 33. | Find the sum to infinity of the geometric progression whose first term is 9 and the second term is $-3.$ (3 marks) | |
| 34. | Given that $2 x-14, x-4$ and $\frac{1}{2} x$ are three consecutive terms of a $GP.$ if $2 x-14$ is the $3^{r d}$ term of a $G P.$ with infinite terms, find the sum to infinity. (3 marks) | |
| 35. | The second term of a $G P.$ is 6 and the sum to infinity is 27. Find the third term of the progression. (5 marks) | |
| 36. | In a GP .$ \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \cdots$, if the difference of the sum to infinity and the sum to the first $n$ terms is less than $\frac{1}{900}$, find the smallest value of $n.$ (5 marks) | |
| 37. | How many terms of the GP. $6,12,24, \ldots$ give a sum of 3066 ? (3 marks) | |
| 38. | If $1+3+3^{2}+3^{3}+\ldots+3^{n}=1093$, find $n.$ (3 marks) | |
| 39. | The first three terms of an $A. P.$ are $x, y, z.$ If these numbers $x, y, z$ are also the first, third and fourth terms of a $G P.$, show that $(2 y-z) z^{2}=y^{3}.$ (5 marks) | |
| 40. | The fourth term of a GP. is 6 and seventh term is $-48.$ Find the eight term of $G P.$ (3 marks) | |
| 41. | In a GP., the fourth term is 18 and seventh term is 486. Find the third term. (3 marks) | |
| 42. | A geometric progression contains seven terms and each term is positive. Given that the first term is 2 and the last term is $\frac{128}{729}$, calculate the middle term. (3 marks) | |
| 43. | The sum of three consecutive terms in a $G P.$ is 42 and their product is 512 , find these terms. (5 marks) | |
| 44. | In a $G P.$ of positive terms, the sum of the first 8 terms is 17 times the sum of its first 4 terms and the fourth term exceeds the second term by 18. Find the first term, the common ratio and the sum to the first 6 terms of the G.P. (5 marks) |
Answer (2012)
| $\quad$ | $\,$ | |
|---|---|---|
| 1. | 11 | |
| 2. | 7 | |
| 3. | 520 | |
| 4. | a=2, d=2 | |
| 5. | 378 | |
| 6. | $d=8 ; 935 $ | |
| 7. | $n=25 ; d=4$ | |
| 8. | $a=1 ; d=2$ | |
| 9. | 5 | |
| 10. | $4 +7+10+13+16+19+22$ | |
| 11. | 31 | |
| 12. | 16 | |
| 13. | 6 | |
| 14. | $5,7,9, \ldots $ | |
| 15. | Show | |
| 16. | $-10 n$ | |
| 17. | $1: 2$ | |
| 18. | Show | |
| 19. | 10 | |
| 20. | $2 n+4 $ | |
| 21. | $2.5,5.5,8.5,11.5 \text { (or) } 11.5,8.5,5.5,2.5$ | |
| 22. | $p=-\frac{1}{2}$ (or) $p=\frac{7}{2} ;n =13$ | |
| 23. | Show | |
| 24. | $ t=7(\text { or) } t=-3 $ | |
| 25. | $q=2 p, r=2$ | |
| 26. | $a=3, r=2$ | |
| 27. | 8 cm, 12 cm | |
| 28. | $\frac{27}{4}$ | |
| 29. | $\frac{729}{64}$ | |
| 30. | $\frac{2}{3} ;189$ | |
| 31. | $\frac{728}{81}$ (or) $\frac{665}{81}$ | |
| 32. | $a=5, b=8, c=12$ (or) $a=17, b=32, c=-24$ | |
| 33. | $\frac{27}{4}$ | |
| 34. | $\frac{243}{2}$ | |
| 35. | 2 (or) 4 | |
| 36. | 5 | |
| 37. | $n=9$ | |
| 38. | 6 | |
| 39. | Show | |
| 40. | 96 | |
| 41. | 6 | |
| 42. | $\frac{16}{27}$ | |
| 43. | $2,8,32$ (or) $32,8,2$ | |
| 44. | $a=3, r=2 ; 183$ |
Group (2011)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 1. | Write down the first four terms of the sequence defined by $u_{n}=3 n-4.$ Which term of the sequence is 74 ? $\mbox{ (3 marks)}$ | |
| 2. | Write down the first four terms of the sequence defined by $u_{n}=4 n-1.$ Which term of the sequence is 191 ? $\mbox{ (3 marks)}$ | |
| 3. | In an A.P.the fifth term is 15 and the ninth term is 31.Find the first term and the 20 -th term of that A.P. $\mbox{ (3 marks)}$ | |
| 4. | In an A.P.the $6^{\text {th }}$ term is 22 and the $10^{\text {th }}$ term is $34.$ Find the $n^{\text {th }}$ term. $\mbox{ (3 marks)}$ | |
| 5. | If $9, x, y, 75$ is an A.P., find the values of $x$ and $y$. $\mbox{ (3 marks)}$ | |
| 6. | Which term of A.P.$3,8,13,18, \ldots$ is 48 ? $\mbox{ (3 marks)}$ | |
| 7. | Which term of the A.P.$10,11 \frac{1}{2}, 13,14 \frac{1}{2}, \ldots$ is $89 \frac{1}{2}$ ? $\mbox{ (3 marks)}$ | |
| 8. | The sum of the first five consecutive terms of an A.P.is $110.$ Find the middle term. $\mbox{ (3 marks)}$ | |
| 9. | If $S_{5}$ is the sum of first 5 terms of A.P., $S^{*}$ is the sum of next 5 terms and $S^{*}-S_{5}=75$, find the common difference of the series. $\mbox{ (3 marks)}$ | |
| 10. | The third and the sixth terms of an A.P.are 13 and 22 respectively, find the sum of the first $n$ terms in terms of $n$. $\mbox{ (5 marks)}$ | |
| 11. | If $k$ is a positive integer, show that the sum of the A.P.$3 k+2,3 k+5,3 k+8, \ldots$, $3 k+44$ is divisible by $5.$ $\mbox{ (5 marks)}$ | |
| 12. | The $4^{\text {th }}$ term of an A.P.is 18 and the sum of the first ten terms is $225.$ Find the $1^{\text {st }}$ term and common difference.Find also $u_{7}$. $\mbox{ (5 marks)}$ | |
| 13. | Find the sum of first 20 terms of the A.P.$2,5,8, \ldots$ Find also the sum of the terms between the $25^{\text {th }}$ term and the $40^{\text {th }}$ term of that A.P. $\mbox{ (5 marks)}$ | |
| 14. | An arithmetic progression contains 20 terms.Given that the eighth term is 25 , and that the sum of the last 8 terms is 404 , calculate the sum of the first 8 terms. $\mbox{ (5 marks)}$ | |
| 15. | The sum of the first four terms of A.P.is 38.The sum of their squares is 406.Find the third term and fourth term. $\mbox{ (5 marks)}$ | |
| 16. | The sum of the first 6 terms of an A.P.is 96.The sum of the first 10 terms is onethird of the sum of the first 20 terms.Calculate the first term and the tenth term. $\mbox{ (5 marks)}$ | |
| 17. | The sum of the first $n$ terms of an A.P.is 30.The common difference is 3 and the sum of the first $2 n$ terms is 108.Find the first term. $\mbox{ (5 marks)}$ | |
| 18. | In an A.P.the sum of the first $n$ 'terms is 21 and the sum of the next $n$ terms is 57.If common difference is 4 , find the first term. $\mbox{ (5 marks)}$ | |
| 19. | The sum of the first 4 terms of an A.P.is 30.The sum of the squares of the $2^{\text {nd }}$ and $3^{\text {rd }}$ terms is 117.Find the first four terms and the $n^{\text {th }}$ term of that A.P. $\mbox{ (5 marks)}$ | |
| 20. | In a G.P.the fourth term is 12 and the seventh term is 96.Find the first term and the ninth term of that G.P. $\mbox{ (3 marks)}$ | |
| 21. | Write down the next two terms of the sequence $\sqrt{2}, \sqrt{10}, 5 \sqrt{2}, 5 \sqrt{10}, \ldots$ and determine the $n^{\text {th }}$ term of the sequence. $\mbox{ (3 marks)}$ | |
| 22. | In a G.P.the $3^{\text {rd }}$ term is 20 and $7^{\text {th }}$ term is 320.Find $n^{\text {th }}$ term. $\mbox{ (3 marks)}$ | |
| 23. | If $3, x, y, 375$ is a G.P., find the values of $x$ and $y$. $\mbox{ (3 marks)}$ | |
| 24. | If $(2 a-1),(4 a+1),(15 a-3), \ldots$ is a G.P., find the values of $a$ and common ratios. $\mbox{ (3 marks)}$ | |
| 25. | Which term of the G.P.$x^{5}, x^{4} y, x^{3} y^{2}, x^{2} y^{3}, \ldots$ is $\frac{y^{10}}{x^{5}}$ ? $\mbox{ (3 marks)}$ | |
| 26. | The product of the first five consecutive terms of a G.P.is $1.$ Find the middle term. $\mbox{ (3 marks)}$ | |
| 27. | Determine whether the sum to infinity of the G.P.$3,0.3,0.03, \ldots$ exists or not; and find it if it exists. $\mbox{ (3 marks)}$ | |
| 28. | In an infinite GP.the common ratio is $-\frac{1}{2}$, the sum to infinity is 6, find the $2^{\text {nd }}$ term and $3^{\text {rd }}$ term. $\mbox{ (3 marks)}$ | |
| 29. | In a G.P., $u_{4}=7$ and $u_{7}=4$.Find the sum to 9 terms of G.P. $\mbox{ (5 marks)}$ | |
| 30. | The sum of the first three terms of a G.P.is 27 and the sum of the $4^{\text {th }}, 5^{\text {th }}$ and $6^{\text {th }}$ terms is $-1.$ Find the sum to infinity of the G.P. $\mbox{ (5 marks)}$ | |
| 31. | The sum of the first three terms of a G.P.is 63 and the sum of the $4^{\text {th }}, 5^{\text {th }}$ and $6^{\text {th }}$ terms is $-\frac{7}{3}$.Find the sum to infinity of the G.P. $\mbox{ (5 marks)}$ | |
| 32. | The first term of a G.P.is 1 more than the second term and its sum to infinity is 4.Find the values of the common ratio and the fifth term. $\mbox{ (5 marks)}$ | |
| 33. | The sum of the first five terms of an infinite G.P.of positive terms is 124.The sum of the terms from the $5^{\text {th }}$ term to the $9^{\text {th }}$ term both inclusive is $7 \frac{3}{4}$.Find the first term, common ratio and the sum to infinity of that G.P. $\mbox{ (5 marks)}$ | |
| 34. | In a G.P.the product of three consecutive terms is 512.When 8 is added to the first term and 6 to the second, then the terms form an A.P.Find the terms of that G.P. $\mbox{ (5 marks)}$ | |
| 35. | Evaluate $(b+5)+\left(b^{2}+5\right)+\left(b^{3}+5\right)+\ldots$ to 38 terms. $\mbox{ (5 marks)}$ | |
| 36. | The G.M.between $x$ and $y+1$ is 12.The A.M.between $x-1$ and $y$ is also 12.Find the values of $x$ and $y$. $\mbox{ (5 marks)}$ | |
| 37. | The ratio of two numbers is $9: 1$.If the sum of the arithmetic mean and geometric mean between the two numbers is 96 , find the two numbers. $\mbox{ (5 marks)}$ |
Answer (2011)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 1. | $-1,2,5,8, u_{26}$ | |
| 2. | $3,7,11,15, u_{48}$ | |
| 3. | $a=-1, u_{20}=75$ | |
| 4. | $u_{n}=3 n+4$ | |
| 5. | $x=31, y=53$ | |
| 6. | $u_{10}$ | |
| 7. | $u_{54}$ | |
| 8. | 22 | |
| 9. | 3 | |
| 10. | $\frac{n}{2}(3 n+11)$ | |
| 11. | Show | |
| 12. | $a=9, d=3, u_{7}=27$ | |
| 13. | 6,101,351 | |
| 14. | 116 | |
| 15. | 11,14 (or) 8,5 | |
| 16. | $a=11, u_{10}=29$ | |
| 17. | $a=3$ | |
| 18. | $a=3$ | |
| 19. | 3,6,9,12 ;$ u_{n}=3 n ; 12,9,6,3 ; u_{n}=15-3 n$ | |
| 20. | $a=\frac{3}{2}, u_{9}=384$ | |
| 21. | $25 \sqrt{2}, 25 \sqrt{10}, u_{n}=\sqrt{2}(\sqrt{5})^{n-1}$ | |
| 22. | $u_{n}=5(\pm 2)^{n-1}$ | |
| 23. | $x=15, y=75$ | |
| 24. | $a=\frac{1}{14}, r=-\frac{3}{2}$ (or) $a=2,r=3$ | |
| 25. | $u_{11}$ | |
| 26. | 1 | |
| 27. | $S=\frac{10}{3}$ | |
| 28. | $-\frac{9}{2}, \frac{9}{4}$ | |
| 29. | $\frac{\frac{279}{28}}{1-\sqrt[3]{\frac{4}{7}}}$ | |
| 30. | $\frac{729}{28}$ | |
| 31. | $\frac{1701}{28}$ | |
| 32. | $r=\frac{1}{2}, \frac{1}{8}$ | |
| 33. | $a=64, r=\frac{1}{2}, S=128$ | |
| 34. | $16,8,4 ; 4,8,16$ | |
| 35. | $\frac{b\left(1-b^{38}\right)}{1-b}+190$ | |
| 36. | $x=8, y=17$ (or) $x=18, y=7$. | |
| 37. | $x=108, y=12$ |
Group (2010)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 1. | The sum to first $n$ terms of a series is $S_{n}=3 n+4 n^{2}$.Find $u_{1}, u_{2}$ and $u_{n}$. $\text{ (3 marks)}$ | |
| 2. | If $a, b, 72,(a+b)$ are consecutive terms of an A.P. , find the values of $a$ and $b$. $\text{ (3 marks)}$ | |
| 3. | If the third and sixth terms of an A.P are 11 and 23 respectively, find the tenth term of the A.P. $\text{ (3 marks)}$ | |
| 4. | In an A.P. the sixth term is 35 and the tenth term is 51. Find the first term and the common difference.$\text{ (3 marks)}$ | |
| 5. | The $6^{\text {th }}$ term of an A.P. is 21 and the sum of the first 17 terms is 0. Find the common difference of the A.P. $\text{ (3 marks)}$ | |
| 6. | If the third term of an A.P. is 23 and the sum of the first 6 terms of the A.P. is 165 , then find the tenth term of the A.P. $\text{ (3 marks)}$ | |
| 7. | The first term of an A.P. is 3 , its $n^{\text {h }}$ term is 23. If the sum of the first $n$ terms is 143 , find $n$ and the common difference. $\text{ (3 marks)}$ | |
| 8. | The last term of an A.P. of 20 terms is 190 and the common difference is 5. Find the sum of the progression. $\text{ (3 marks)}$ | |
| 9. | The three angles of a triangle form an $A. P. $ If the largest angle is twice the smallest angle, find three angles of that triangle.$\text{ (3 marks)}$ | |
| 10. | Insert 2 A.M. between 12 and 96. $\text{ (3 marks)}$ | |
| 11. | The ninth term of an A.P. is 26 and the sum to the first 11 terms is 187. Find the $n^{\text {th }}$ term of that A.P. $\text{ (5 marks)}$ | |
| 12. | The sum of first six terms of an A.P. is $55.5$ and the sum of the next six terms is 145.5Find the common difference of the A.P. and the first term. $\text{ (5 marks)}$ | |
| 13. | The ninth term of an A.P. is 42 and the sum from the sixth term to the tenth term of the A.P. exceeds the sum of the first 5 terms of the $A$.P by 100. Find the sum of the first 15 terms of the $A. P$. $\text{ (5 marks)}$ | |
| 14. | The seventh term of an A.P. is 15 and the sum from sixth term to the tenth term of the A.P. exceeds the sum of the first 5 terms of the A.P. by 50. Find the sum of the first 10 terms of the A.P. $\text{ (5 marks)}$ | |
| 15. | In an A.P. $25,19,13, \ldots$, find the $11^{\text {th }}$ term and the sum of the first 11 terms of the A.P. .Find also the sum from $12^{\text {th }}$ term to $30^{\text {th }}$ term. $\text{ (5 marks)}$ | |
| 16. | The sum of the first 4 terms of an A.P. is 24 and the sum of their squares is 204. Find the first 4 terms. $\text{ (5 marks)}$ | |
| 17. | The sum of the first four terms of an $A$. P. is 26 and the sum of their squares is 214. Find the first four terms. $\text{ (5 marks)}$ | |
| 18. | Find the sum of all three-digit natural numbers which are divisible by 3. $\text{ (5 marks)}$ | |
| 19. | Find the sum of all three-digit natural numbers which are divisible by 4. $\text{ (5 marks)}$ | |
| 20. | Find the sum of all three-digit natural numbers which are divisible by 5. $\text{ (5 marks)}$ | |
| 21. | The sum of the squares of three consecutive numbers in an $A. P$.equals 165. The sum of the numbers is 21. Find the numbers. $\text{ (5 marks)}$ | |
| 22. | Given that $x+1, x+5$ and $2 x+4$ are three consecutive terms of a $G P$.Find the value of $x$. $\text{ (3 marks)}$ | |
| 23. | If the second and fifth terms of a $G P$.are 6 and 48 respectively, find the eighth term of the $G P$. $\text{ (3 marks)}$ | |
| 24. | If the third and sixth terms of a $G P$.are 40 and 320 respectively, then find the eighth term of the G.P. $\text{ (3 marks)}$ | |
| 25. | The fifth term of a $G P$.of positive terms is 48 and the seventh term is 192. Find the first term. $\text{ (3 marks)}$ | |
| 26. | The $4^{\text {th }}$ term of a GP. is 1 and the $8^{\text {th }}$ term is $\frac{1}{256}$.Find the first term of the $G P$. $\text{ (3 marks)}$ | |
| 27. | The first term and fourth term of a $G P$.are 192 and 3 respectively.Find the sum to infinity. $\text{ (3 marks)}$ | |
| 28. | If the sum to infinity of a $G P. $, is twice the first term and the fifth term is $\frac{1}{16}$, find the first term. $\text{ (3 marks)}$ | |
| 29. | Find $n$ if $1+2+2^{2}+\ldots+2^{n}=511$. $\text{ (3 marks)}$ | |
| 30. | If $4, x, y$ is an $A. P. $ and $\frac{1}{4}, 4, y$ is a $G P$, then find the values of $x$ and $y$. $\text{ (3 marks)}$ | |
| 31. | Insert 2 G.M. between 28 and 224.$\text{ (3 marks)}$ | |
| 32. | The $3^{\text {rd }}$ term of a $G P$.is $-8$ and the $8^{\text {th }}$ term is 256. Find the sum of the first 6 terms. $\text{ (5 marks)}$ | |
| 33. | In a $G P$., the first term exceeds the third term by 72 and the sum of the second and third terms is 36. Find the first term. $\text{ (5 marks)}$ | |
| 34. | The first term of a $G P$.is $a$ and common ratio $r$ is positive.If the sum of second and the third term is $\frac{10 a}{9}$ and the sum of the first 4 terms is 65 , then find $a$ and $r$. $\text{ (5 marks)}$ | |
| 35. | The product of the first three terms of a GP. is 27. The ratio of the sum of the first 4 terms of the $G P$.and the sum of the first 2 terms is $10: 1$.Find the seventh term of the $G P$. $\text{ (5 marks)}$ | |
| 36. | The product of the first three terms of a $G P$.is 27. The ratio of the sum of the first 6 terms of the $G P$.and the sum of the first 3 terms of the G.P. is $9: 1$.Find the tenth term of the $G P$. | |
| 37. | An infinite geometric series has a finite sum.Given that the first term is 18 and that the sum of the first 3 terms is 38 , calculate the value of the common ratio and the sum to infinity. $\text{ (5 marks)}$ | |
| 38. | In a $G P$.the sum of the first three terms is $4 \frac{2}{9}$ and the sum to infinity is $6. $ Find the sixth terms of the $G P$. $\text{ (5 marks)}$ | |
| 39. | In an infinite $G P$., the sum to infinity is 16 and the sum to the first 3 terms is 14. Find the fifth term of the $G P$. | |
| 40. | Find $n$, if $2+2^{2}+2^{3}+\ldots+2^{n}=510$. $\text{ (5 marks)}$ | |
| 41. | The lengths of the sides of a triangle form a GP. The length of the longest side exceeds that of the shortest side by $9 \mathrm{~cm}$.The perimeter of the triangle is $61 \mathrm{~cm}$.Find the length of each side. $\text{ (5 marks)}$ | |
| 42. | If $x, y, z$ is an $A,P$.and $a, b, c$ is an $GP$., show that $a^{y-z} b^{z-x} c^{x-y}=1$. $\text{ (5 marks)}$ |
Answer (2010)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 1. | $7 ; 15 ; 8 n-1$ | |
| 2. | $36 ; 54$ | |
| 3. | 39 | |
| 4. | 15 ; 4 | |
| 5. | $-7 $ | |
| 6. | 86 | |
| 7. | 11 ; 2 | |
| 8. | 2850 | |
| 9. | $40^{\circ}, 60^{\circ}, 80^{\circ} $ | |
| 10. | 12,40,68,96 | |
| 11. | $ U n=3 n-1 $ | |
| 12. | $ d=2.5, a=3 $ | |
| 13. | 570 | |
| 14. | 120 | |
| 15. | $U_{11}=-35;S_{11}=-55;-1805$ | |
| 16. | $6-3\sqrt 3, 6-\sqrt 3, 6+\sqrt 3, 6+3\sqrt 3;$ $6+3\sqrt 3, 6+\sqrt 3,6-\sqrt 3, 6-3\sqrt 3$ | |
| 17. | 2,5,8,11; 11,8,5,2 | |
| 18. | 165150 | |
| 19. | 123300 | |
| 20. | 98550 | |
| 21. | 4,7,10;10,7,4 | |
| 22. | $-3$ or 7 | |
| 23. | 384 | |
| 24. | 1280 | |
| 25. | 3 | |
| 26. | $ \pm 64 $ | |
| 27. | 256 | |
| 28. | 1 | |
| 29. | 8 | |
| 30. | 34,64 | |
| 31. | 28,56,112,224 | |
| 32. | 42 | |
| 33. | 81 | |
| 34. | $a=27,r=\frac 23$ | |
| 35. | $\pm 729$ | |
| 36. | 768 | |
| 37. | $r=\frac{2}{3} ; s=54$ | |
| 38. | $\frac{64}{243}$ | |
| 39. | $\frac{1}{2} $ | |
| 40. | 8 | |
| 41. | 16,20,25 | |
| 42. | Show |
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