Arithmetic & Geometric Series (Edexcel IGCSE Further Pure Maths): Exam Questions

The $n$th term of a geometric series is $3{\mathrm{e}}^{\left(1–2n\right)}$

Find the sum to infinity of this series.

Give your answer in the form $\frac{a\mathrm{e}}{{\mathrm{e}}^b-1}$ where $a$and $b$ are integers to be found.

Expand $\frac{3}{\sqrt{1-2x}}$ in ascending powers of $x$ up to and including the term in $x^3$

and simplifying each term as far as possible.

Write down the range of values of $x$ for which this expansion is valid.

Show that $\frac{3}{\sqrt{0.9}}=\sqrt{10}$

Express $\frac{1}{\sqrt{10}-3}$ in the form $a\sqrt{10}+b$ , where $a$ and $b$ are integers.

Hence, using your expansion with a suitable value for $x$, obtain an approximation to $5$ decimal places of $\frac{1}{\sqrt{10}-3}$

Show that $\sum _{r=1}^n\left(5r-3\right)=\frac{n}{2}\left(5n-1\right)$

Hence, or otherwise, evaluate $\sum _{r=31}^{60}\left(5r-3\right)$

Given that $\sum _{r=1}^n\left(5r-3\right)=3783$

find the value of $n$

The sum of the first and second terms of a geometric series $G$ is $400$

The sum of the second and third terms of $G$ is $100$

Show that the common ratio of $G$ is $\frac{1}{4}$

Show that the first term of $G$ is $320$

Find the sum to infinity of $G$

The sum to $n$ terms of $G$ is $S_n$

Find, using logarithms, the least value of $n$ such that

$S_n>426.6$

The sum of the fifth, sixth and seventh terms of an arithmetic series$A$ is nine times the sum of the first and second terms.
The third term of$A$ is 12

Find the first term and common difference of $A$

The $n$th term of $A$ is $u_n$

Find the value of $\sum _{r=15}^{60}{u}_r$

The sum to $n$ terms of $A$ is $S_n$

Given that $2S_n-5u_n=10$

find the value of $n$

Use the factor theorem to show that $\left(4x-1\right)$ is a factor of

$\mathrm{f}\left(x\right)=64x^3-64x^2+3$

Use the factor theorem to show that $\left(4x-1\right)$ is a factor of

$\mathrm{f}\left(x\right)=64x^3-64x^2+3$

Hence, or otherwise, find the exact roots of the equation

$\mathrm{f}\left(x\right)=0$

A geometric series $G$ has first term $a$ and common ratio $r$
The third term of $G$ is 9 and the sum to infinity of $G$ is 192

Show that $64r^3-64r^2+3=0$

A geometric series $G$ has first term $a$ and common ratio $r$
The third term of $G$ is 9 and the sum to infinity of $G$ is 192

Given that $r$ is a rational number

write down the value of $r$

A geometric series $G$ has first term $a$ and common ratio $r$
The third term of $G$ is 9 and the sum to infinity of $G$ is 192

show that $a$ =144

The sum to $n$ terms of $G$ is $S_n$

Using logarithms, find the least value of $n$ such that $S_n>191.9$

Show that $\sum _{r=1}^n(3r+2)=\frac{n}{2}(3n+7)$

Hence, or otherwise, evaluate $\sum _{r=10}^{40}(3r+2)$

The $n$th term of a convergent geometric series is $8^{(1-2n)}$

Find the sum to infinity of the series.

Give your answer in the form $\frac{p}{q}$where $p$ and $q$ are integers to be found.

A geometric series $G$ with common ratio $r$, has first term 16 and third term $\frac{2704}{625}$

Find the two possible values of $r$

Given that $r>0$

find the sum to infinity of $G$

The sum to $n$ terms of $G$ is greater than 33

Find, using logarithms, the least possible value of $n$
Show your working clearly.

The sum to $n$ terms of an arithmetic series $A$ is $S_n$

The sum of the first four terms of $A$is 42 and the fifth term of $A$is 23

Show that $S_n=\sum _{r=1}^n(Pr-Q)$ where $P$ and $Q$ are prime numbers.

$S_{2n}-3U_n=1062$ where $U_n$ is the $n$th term of $A$

Find the value of $n$

The $n$th term of an arithmetic series is $a_n$ where

$a_{10}+a_{11}+a_{12}=129$ and $a_{19}+a_{20}+a_{21}=237$

Find $a_1$

The $n\mathrm{th}$ term of a geometric series $G$ is $U_n$

The first three terms of $G$ are given by

$U_1=q(4p+1)U_2=q(2p+3)U_3=q(2p-3)$

Find the possible values of $p$

Given that $G$ is convergent with sum to infinity 250

find the value of $q$