Geometric Progressions (Cambridge International A-Level Mathematics): Exam Questions

Identify which of the following are geometric progressions. For those that are, write down the first term and the common ratio.

(i) $3 , 8 , 13 , 18 , \dots$

(ii) $5 , 15 , 45 , 135 , \dots$

(iii) $5 , - 10 , 20 , - 40 , \dots$

(iv) $\frac{1}{3} , \frac{1}{6} , \frac{1}{12} , \dots$

Write down a formula for the $n$th term of each of the following geometric progressions

(i) $3 , 12 , 48 , 192 , \dots$

(ii) First term $a = 5$, common ratio $r = - 2$

(iii) $a = 16 , r = \frac{1}{2}$

Find the 5th and 10th terms in each of the following geometric progressions

(i) $u_{n} = 2 ( 3 )^{n}$

(ii) $u_{n} = 10 000 ( 1 . 02 )^{n}$

(iii) $u_{n} = 3^{-n}$

The first term of a geometric progression is 6.

The sum to infinity of the progression is 8.

Show that the common ratio of the progression is 0.25.

For a geometric progression with first term 6 and common ratio 0.25, briefly explain why the sum to infinity will exist.

A geometric progression has first term 5 and common ratio $\frac{3}{2}$.

Find the sum of the first 12 terms of the progression. Give your answer correct to the nearest whole number.

A different geometric progression has first term 4 and common ratio $\frac{1}{8}$.

Find the sum to infinity of the progression.

The first term of a geometric progression is 2.

The sixth term of the progression is 486.

The sum of the first $n$ terms of the progression is 177 146.

Find the common ratio of the progression.

Hence show that $3^{n} = 177147$.

The first three terms of a geometric progression are $k ( k + 1 )$, $k ( k + 1 )^{2}$ and $k ( k + 1 )^{3}$, where $k$ is a constant.

Write down a formula for the $n$th term of the progression, in terms of $k$.

Given that the progression is convergent, show that the sum to infinity is $- ( k + 1 )$.

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

The $k$th term of a geometric progression is given by $u_{k} = 5 \times 2^{k-1}$.

Write down the first five terms of the progression.

Calculate the sum of the first five terms of the progression.

The third and sixth terms of a geometric progression are 10 and 270 respectively.

Find the first term and the common ratio of the progression.

A different geometric progression is such that the twelfth term is equal to 16 times the eighth term.

Find the possible values of the common ratio.

The first three terms of a geometric progression are $x^{2}$, $4 x$ and $x + 14$ respectively, where $x > 0$.

Show that $x^{3} - 2 x^{2} = 0$.

Find the value of the fifteenth term of the progression.

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

The sum of the first two terms of a geometric progression is 9.31.

The sum of the first four terms of the same progression is 11.02.

The common ratio of the progression is $r$.

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

Hence find the two possible values of $r$.

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

A different geometric progression has first term $b$ and common ratio 3.

The sum of the first three terms of both progressions is the same.

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

The first three terms of a geometric progression are $( k - 3 )$, $k$ and $( 2 k + 8 )$, where $k$ is a positive constant.

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

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

Find the common ratio of the progression.

Find the sum of the first 12 terms of the progression.

The first three terms of a geometric progression are $- 1$, $3 x$ and $- 9 x^{2}$.

Given that the progression is convergent, write down an inequality that the common ratio of the progression must satisfy, and hence find the range of possible values of $x$.

Find an expression for the sum to infinity of the progression, in terms of $x$.

A geometric progression has first term 64 and sum to infinity 384.

Show that the common ratio, $r$, of the progression is $\frac{5}{6}$.

Find the difference between the ninth and tenth terms of the progression. Give your answer correct to 3 significant figures.

Calculate the sum of the first eight terms of the progression. Give your answer correct to 3 significant figures.

Given that the sum of the first $k$ terms of the progression is greater than 380, show that

\left(\dfrac{5}{6}\right)^{k} < \dfrac{1}{96}

The $k$th term of a geometric progression is given by u_{k} = 162\left(\dfrac{1}{3}\right)^{k}.

Calculate the sum of the first nine terms of the progression, giving your answer as an exact value.

Calculate the sum to infinity of the progression starting from the tenth term, giving your answer as an exact value.

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

Show that the sum of the first ten terms of the progression is equal to ka\left(\sqrt{5}+1\right), where $k$ is a positive integer to be determined.

The second and fifth terms of a geometric progression are 13.44 and 5.67 respectively.

The progression has first term $a$ and common ratio $r$.

By first determining the values of $a$ and $r$, calculate the sum to infinity of the progression.

Calculate the difference between the sum to infinity of the progression and the sum of its first 20 terms. Give your answer correct to 2 decimal places.

A geometric progression has first term 9 and the sum of its first three terms is 19.

The common ratio of the progression is $r$.

Show that $9 r^{2} + 9 r - 10 = 0$.

Find the two possible values of $r$.

Given that the progression is convergent, find the sum to infinity of the progression.

The first three terms of a geometric progression are $x + 12$, $3 x$ and $x^{2}$ respectively, where $x$ is a non-zero real number.

Find the value of the 102^nd term of the progression.

The sum of the first three terms of a geometric progression is 8.75.

The sum of the first six terms of the same progression is 13.23.

Find the common ratio of the progression.

The first three terms of a geometric progression are $( 2 k + 3 )$, $k$ and $( k - 2 )$, where $k$ is a negative constant.

Find the value of $k$.

Find the sum of the first 12 terms of the progression.

The $k$th term of a geometric progression is given by u_{k} = 2401\left(\dfrac{2}{7}\right)^{k}.

Calculate the sum to infinity of the progression starting with the seventh term, giving your answer as an exact value.

A second progression has $k$th term $v_{k} = u_{k+4}$, where $u_{k}$ is defined as above.

Calculate the sum to infinity of this progression, giving your answer as an exact value.

The first three terms of a geometric progression are $x + 11$, $5 x$ and $3 x^{2}$ respectively, where $x$ is a non-zero real number.

Find the value of the sixth term of the progression, giving your answer as a fraction.

The sum of the first four terms of a geometric progression is 27.2.

The sum of the first eight terms of the same progression is 164.9.

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

A geometric progression has first term $a$, and its terms are connected by the relationship $u_{n+4} = 9 u_{n}$ for all $n \geq 1$.

Given that all the terms of the progression are positive, show that the sum of the first twelve terms of the progression may be written in the form

S_{12} = ka\left(\sqrt{n} + 1\right)

where $k$ and $n$ are positive integers and $\sqrt{n}$ is a surd.

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

Find the possible values of $k$.

Given that the progression is convergent, find the sum to infinity of the progression.

The second and third terms of a geometric progression are $( x - 1 )$ and $( x^{2} - 1 )$, where $x$ is a real number not equal to $1$ or $- 1$.

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

Given that the sum to infinity of the progression is $- 6$, find the two possible values of $x$.

The geometric progression $S$ is defined by $S = u_{1} + u_{2} + u_{3} + \dots + u_{n} + \dots$, where $u_{n}$ denotes the $n$th term of the progression. The sum to infinity of the progression exists and is denoted by $S_{\infty}$. The first term of the progression, $u_{1}$, is equal to $a$, and the common ratio of the progression is $r$.

A different progression $T = u_{1}^{2} + u_{2}^{2} + u_{3}^{2} + \dots + u_{n}^{2} + \dots$ is formed by squaring all the terms of the progression $S$ above.

Show that $T$ is also a geometric progression, and that its sum to infinity also exists.

The sum to infinity of the progression $T$ 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 $k \geq 1$. Comment on what this shows about the relationship between the terms of the two progressions.

The $k$th term of a geometric progression is given by $u_{k} = ( - 2 )^{k-1}$.

Calculate the sum of the eleventh through twenty-third terms of the sequence whose $k$th term is given by $v_{k} = u_{k} + \sqrt{13}$, where $u_{k}$ is defined as above. Give your answer as an exact value.