Geometric Sequences & Series (Edexcel A Level Maths: Pure): Exam Questions

In this question you should show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A company made a profit of £20 000 in its first year of trading, Year 1.

A model for future trading predicts that the yearly profit will increase by 8% each year, so that the yearly profits will form a geometric sequence.

According to the model, show that the profit for Year 3 will be £23 328.

According to the model, find the first year when the yearly profit will exceed £65 000.

According to the model, find the total profit for the first 20 years of trading, giving your answer to the nearest £1 000.

A car has six forward gears.

The fastest speed of the car

• in 1^st gear is 28 km h^-1

• in 6^th gear is 115 km h^-1

Given that the fastest speed of the car in successive gears is modelled by a geometric sequence,

find the fastest speed of the car in 5^th gear.

A geometric sequence is defined as follows

• The first term is 2

• The sixth term is 486

Find the common ratio.

The sum of the first $n$ terms is $177146$ .

Show that 

$3^n=177147$

Hence find the value of $n$.

In a geometric sequence

• the 3^rd term is 10

• the 6^th term is 270

Find the first term and the common ratio.

In a different geometric sequence, the 12^th term is 16 times greater than the 8^th term.

Find the possible values of the common ratio.

In a geometric series,

• the first term is 19

• the common ratio is $\frac{2}{3}$

• the sum of the first $k$ terms of the series is greater than 56

(i) Show that 

$k>\frac{\log \left({\displaystyle \frac{1}{57}}\right)}{\log \left({\displaystyle \frac{2}{3}}\right)}$

(ii) Hence find the smallest possible value of $k$.

The sum of the first two terms in a geometric series is 9.31

The sum of the first four terms in the same geometric series is 11.02

The common ratio of the geometric series is $r$ where $r\ne 1$

 Show that

$\frac{1-r^4}{1-r^2}=\frac{58}{49}$

Hence find the possible values of $r$.

A geometric series has first term $a$ and common ratio $r$. Given that the second term is 12 and the sum to infinity is 64, find two possible values of $r$.

The first three terms in a geometric sequence are $(k-3)$, $k$, $(2k+8)$, where $k>0$ is a constant.

(i) Show that $k^2+2k-24=0$

(ii) Hence find the value of $k$.

Find the common ratio of the sequence.

 Find the sum of the first 12 terms.

Given that the geometric series 

$-1+3x-9x^2+27x^3+\dots$ 

is convergent, find the range of possible values of $x.$

Assuming the series is convergent, find an expression for the sum to infinity of the series in terms of $x$.

The first term of a geometric series is $a$, and its common ratio is 5. 

A different geometric series has a first term of $b$ and a common ratio of 3. 

For both series, the sum of the first three terms are equal.

Find the value of $\frac{a}{b}$, giving your answer as a fraction in simplest form.

A geometric sequence is given by

$k\left(k+1\right),k{\left(k+1\right)}^2,k{\left(k+1\right)}^3,k{\left(k+1\right)}^4,...$

where $k$ is a constant such that $|k+1|<1$.

Find, in terms of $k$ and $n$, a formula for the $n$^th term of the sequence.

The sequence is part (a) is used to form a series.

Show that the sum to infinity of the series is

$-(k+1)$

Given that the sum to infinity is $-\frac{1}{4}$, find the value of $k$.

Find the value of

$\sum _{r=1}^5$ $3\left(2^r\right)$

Find the value of

         $\sum _{r=4}^8$ ${\left(-1\right)}^r\left(2^r\right)$

A geometric series has first term of $a$ and common ratio of $\sqrt{5}$.

Show that the sum of the first ten terms of the series is

$ka(\sqrt{5}+1)$

where $k$ is a positive integer to be found.

The first three terms of a geometric sequence are given by

$x^2,4x,\left(x+14\right)$

where $x>0.$

Show that

$x^3-2x^2=0$

Find the value of the 15^th term of the sequence.

State, with a reason, whether 8192 is a term in the sequence.

A geometric series is defined as follows:

• The first term is 64

• The sum to infinity is 384

Show that the common ratio is $\frac{5}{6}$.  

Find the difference between the 9^th term and the 10^th term of the series, to 3 significant figures.

Find the sum of the first eight terms in the series, to 3 significant figures.

Given that the sum of the first $k$ terms of the series is greater than 380, find the smallest possible value of $k$.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A geometric series has common ratio $r$ and first term $a$.

Given $r\ne 1$ and $a\ne 0$, prove that

$S_n=\frac{a\left(1-r^n\right)}{1-r}$

Given also that $S_{10}$ is four times $S_5$, find the exact value of $r$.

The first three terms of a geometric sequence are

$3k+412–3kk+16$

where $k$ is a constant.

Show that $k$ satisfies the equation

$3k^2–62k+40=0$

Given that the sequence converges,

(i) find the value of $k$, giving a reason for your answer,

(ii) find the value of $S_{\infty }$

The first three terms of a geometric sequence are given by

$\left(x+12\right),3x,x^2$

where $x$ is a non-zero real number.

Find the value of the 102^nd term in the sequence.

In a geometric sequence,

• the second term is 4

• the common ratio is $r$ where $r>0$

• the 16^th term is 9

Show that $r$ satisfies the equation

$14\ln r+\ln \left(\frac{4}{9}\right)=0$

Find the value of $r$, to 3 significant figures.

A geometric series has first term of 14 and common ratio of  $\frac{99}{100}$.

Given that the sum of the first $k$ terms of the series is less than 1000, find the largest possible value of $k$.

The sum of the first three terms in a geometric series is 8.75

The sum of the first six terms in the same geometric series is 13.23

Find the common ratio of the series.

The first three terms in a geometric sequence are $(2k+3)$, $k$, $(k-2)$, where $k<0$ is a constant.

Find the value of $k$.

Find the sum of the first 12 terms.

In a geometric series,

• the second term is 13.44

• the fifth term is 5.67

Assuming the series is convergent, find the sum to infinity of the series.

Find the difference between the sum to infinity of the series and the sum of the first 20 terms of the series, to 2 decimal places.

In a geometric series

• the first term is 9

• the sum of the first three terms is 19

• the common ratio is $r$ where $r\ne 1$

Show that

$9r^2+9r-10=0$

Find the possible values of $r$.

Given that the series converges, find the sum to infinity of the series.

The first three terms of a geometric sequence are given by

$\left(x+11\right),5x,3x^2$

 where $x$ is a non-zero real number.

Find the 6^th term in the sequence, giving your answer as a fraction.

The sum of the first four terms in a geometric series is 27.2

The sum of the first eight terms in the same geometric series is 164.9

Given that the first term is positive, find the common ratio of the series.

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Given that the first three terms of a geometric series are

$12\cos \theta 5+2\sin \theta$and$6\tan \theta$

show that

$4{\sin }^2\theta -52\sin \theta +25=0$

Given that $\theta$ is an obtuse angle measured in radians, solve the equation in part (a) to find the exact value of $\theta$

Show that the sum to infinity of the series can be expressed in the form

$k\left(1-\sqrt{3}\right)$

where $k$ is a constant to be found.

In a geometric series,

• the second term is 648

• the fifth term is 375

• the sum of the first $k$terms of the series is greater than 4660

Find the smallest value of $k$.

A geometric series has a first term of $a$ and its terms satisfy the relationship 

$u_{n+4}=9u_n$

for all $n\in \mathrm{\mathbb{N}}$.

Given that all the terms in the series are positive, show that the sum of the first twelve terms of the series is

$ka(\sqrt{n}+1)$

where $k$ and $n$ are positive integers to be found.

The first three terms in a geometric series are $(2k+6)$, $k$, $(k-4)$, where $k$ is a constant.

Find the possible values of $k$.

Given that the sum to infinity exists, find the sum to infinity of this series.

In a convergent geometric series

• the second term is $\left(x-1\right)$

• the third term is $\left(x^2-1\right)$

where $x^2\ne 1$.

Find the range of possible values of $x$.

The sum to infinity of the series is $-6$

Find the possible values of  $x$.

The geometric series $S=u_1+u_2+u_3+\dots +u_n+\dots$ is convergent, and the sum to infinity of the series is $S_{\infty }$.  The first term of the series is $a$, and the common ratio is $r$.

A different series $T=u_1^2+u_2^2+u_3^2+\dots +u_n^2+\dots$ is formed by squaring all the terms of the series $S$ above.

Show that $T$ is also a convergent geometric series.

The sum to infinity of series $T$ is $T_{\infty }$.

Express the ratio $\frac{T_{\infty }}{S_{\infty }}$in terms of $a$ and $r$, simplifying your answer as far as possible.

Show that if $T_{\infty }=S_{\infty }$, then 

$u_k^2=u_{2k-1}+u_{2k}$

for all $k\ge 1$. 

Hence describe the relationship between the terms of the two series in the case when $T_{\infty }=S_{\infty }$.