$\def\frac{\dfrac}$
$\def\view{\begin{array}{|l|}\hline\mbox{Arithmetic Progress (AP)}\\ \hline\begin{array}{r|ll}1&d&=u_2-u_1=u_3-u_2=\cdots\\&d&=\mbox{Common Difference}\\&a&=u_1=\mbox{First Term}\\&u_n&=n^{\mbox{th}}\mbox{ Term }\\&S_n&=\mbox{Sum to first $n$ terms}\\&l&=u_n=\mbox{Last term}\\ 2&u_n&=a+(n-1)d \\ 3&S_n&=\frac n2(2a+(n-1)d) \\&&=\frac n2(a+l)\end{array}\\ \hline\mbox{Geometric Progress (GP)}\\ \hline\begin{array}{r|ll}1&r&=\frac{u_2}{u_1}=\frac{u_3}{u_2}=\cdots\\&r&=\mbox{Common ratio}\\&a&=u_1=\mbox{First Term}\\&u_n&=n^{\mbox{th}}\mbox{ Term }\\&S_n&=\mbox{Sum to first n terms}\\ &S&=S_{\infty}=\mbox{Sum to infinity}\\ 2&u_n&=ar^{n-1}\\ 3&S_n&=\frac{a(1-r^n)}{1-r} \\&S&=\frac{a}{1-r}, -1<r<1\end{array}\\ \hline\end{array}}$
1. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 11 / \mathrm{q} 10)$
(a) An arithmetic progression has a second term of 8 and a fourth term of $18 .$ Find the least number of terms for which the sum of this progression is greater than 1560 .
(b) A geometric progression has a sum to infinity of 72 . The sum of the first 3 terms of this progression is $\frac{333}{8}$
(i) Find the value of the common ratio.
(ii) Hence find the value of the first term.
2. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 11 / \mathrm{q} 9)$
(a) An arithmetic progression has a second term of $-14$ and a sum to 21 terms of $84 .$ Find the first term and the 21 st term of this progression.
(b) A geometric progression has a second term of $27 p^{2}$ and a fifth term of $p^{5}$. The common ratio, $r$, is such that $0<r<1$
(i) Find $r$ in terms of $p$.
(ii) Hence find, in terms of $p$, the sum to infinity of the progression.
(iii) Given that the sum to infinity is 81 , find the value of $p$.
3. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 12 / \mathrm{q} 4)$
The 7 th and 10 th terms of an arithmetic progression are 158 and 149 respectively.
(a) Find the common difference and the first term of the progression.
(b) Find the least number of terms of the progression for their sum to be negative. [3]
4. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 13 / \mathrm{q} 8)$
(a) An arithmetic progression has a first term of 7 and a common difference of $0.4$. Find the least number of terms so that the sum of the progression is greater than 300 .
(b) The sum of the first two terms of a geometric progression is 9 and its sum to infinity is 36 . Given that the terms of the progression are positive, find the common ratio.
5. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 22 / \mathrm{q} 10)$
(a) The first 5 terms of a sequence are given below.
$$\begin{array}{ccccc}4 & -2 & 1 & -0.5 & 0.25\end{array}$$
(i) Find the 20 th term of the sequence.
(ii) Explain why the sum to infinity exists for this sequence and find the value of this sum.
(b) The tenth term of an arithmetic progression is 15 times the second term. The sum of the first 6 terms of the progression is $87 .$
(i) Find the common difference of the progression.
(ii) For this progression, the $n$th term is 6990 . Find the value of $n$.
6. (CIE $0606 / 2020 / \mathrm{m} / 22 / \mathrm{q} 13)$
(a) The sum of the first two terms of a geometric progression is 10 and the third term is $9 .$
(i) Find the possible values of the common ratio and the first term.
(ii) Find the sum to infinity of the convergent progression.
(b) In an arithmetic progression, $u_{1}=-10$ and $u_{4}=14$. Find $u_{100}+u_{101}+u_{102}+\ldots+u_{200}$ ' the sum of the 100 th to the 200 th terms of the progression.
7. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 22 / \mathrm{q} 7)$
A geometric progression has a first term of 3 and a second term of $2.4$. For this progression, find
(a) the sum of the first 8 terms, $[3]$
(b) the sum to infinity, $[1]$
(c) the least number of terms for which the sum is greater than $95 \%$ of the sum to infinity.
8. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 23 / \mathrm{q} 10)$
(a) The sum of the first 4 terms of an arithmetic progression is 38 and the sum of the next 4 terms is 86. Find the first term and the common difference.
(b) The third term of a geometric progression is 12 and the sixth term is $-96$. Find the sum of the first 10 terms of this progression.
Answers
1. (a) 25
(b)(i) $\frac{3}{4}$ (ii) $a=18$
2. (a) $a=-16, u_{21}=24$
(b)(i) $r=\frac{p}{3}$
(ii) $S_{\infty}=\frac{243 p}{3-p}$
(iii) $p=\frac{3}{4}$
3. (a) $d=-3, a=176$ (b) 119
4. (a) 26 terms
(b) $r=\frac{\sqrt{3}}{2}$
5. (a)(i) $-7.63 \times 10^{-6}$
(ii) $S_{\infty}=\frac{8}{3}$
(b)(i) $d=7$ (ii) $n=1000$
6. (a)(i) $r=-\frac{3}{5}, \frac{3}{2}$ and $a=25,4$
(ii) $\frac{125}{8}$ (b) 119382
7. (a) $12.5$
(b) 15 (c) 14
8. (a) $a=5, d=3$
(b) $-1023$
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