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

The sum $S_{n}$ of the first $n$ terms of an arithmetic series is given by $S_{n}=2n(n+3)$

Find the first term of the series.

Find the common difference of the series.

The $n$th term of the series is $T_{n}$

Given that $6 S_{( n - 4 )} = 7 T_{( n + 3 )}$ find the value of $n$.

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

Find the sum to infinity of this series.

Give your answer in the form $\frac{ae}{e^{b}-1}$ 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 $\sum_{r = 1}^{n} ( 5 r - 3 ) = \frac{n}{2} ( 5 n - 1 )$

Hence, or otherwise, evaluate $\sum_{r = 31}^{60} ( 5 r - 3 )$

Given that $\sum_{r = 1}^{n} ( 5 r - 3 ) = 3783$

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 $( 4 x - 1 )$ is a factor of

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

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

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

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

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

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 $\sum_{r = 1}^{n} ( 3 r + 2 ) = \frac{n}{2} ( 3 n + 7 )$

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

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 $S_{n} = \sum_{r = 1}^{n} ( P r - Q )$ 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 $q$