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.

The first three terms of a geometric sequence are given by , , and  respectively, where

Show that .

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 sequence has first term 900 and a common ratio  where .  The 18th term of the sequence is 18.

Show that satisfies the equation

Hence or otherwise find the value of correct to 3 significant figures.

A geometric series has first term 19 and common ratio  .

Given that the sum of the first  terms of the series is greater than 56

(i) show that 

(ii) hence find the smallest possible value of .

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

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

The common ratio of the series is .

 Show that .

Hence find the two possible values of .

The first term of a geometric series is , and its common ratio is 5.  A different geometric series has first term  and common ratio 3.  The sum of the first three terms of both series is the same. 

Find the value of  , giving your answer as a fraction in simplest terms.

The first three terms in a geometric series are , , , where  is a constant.

(i) Show that   .

(ii) Hence find the value of .

Find the common ratio, , of this series.

 Find the sum of the first 12 terms in this series.

Given that the geometric series    is convergent  

find the range of possible values of 

find an expression for the sum to infinity, , in terms of .

A convergent geometric series has first term 64, and the sum to infinity of the series is 384.

Show that the common ratio, , of the series is .  

Find the difference between the ninth and tenth terms of the series, giving your answer correct to 3 significant figures.

Calculate the sum of the first eight terms in the series, giving your answer correct to 3 significant figures.

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

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 and first term .

Given and , prove that

Given also that is four times , find the exact value of .

The first three terms of a geometric sequence are

where is a constant.

Show that satisfies the equation

Given that the sequence converges,

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

(ii) find the value of

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

and

show that

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

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

where is a constant to be found.

The first three terms of a geometric sequence are given by , , and  respectively, where  is a non-zero real number.

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

The second term of a geometric sequence is 4, and the sixteenth term of the same sequence is 9.  The common ratio is , where .

Show that  satisfies the equation, .

Hence or otherwise find the value of  correct to 3 significant figures.

A geometric series has first term 14 and common ratio  .

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

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

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

Find the common ratio, , of the series.

A geometric series has first term  and common ratio .

Show that the sum of the first ten terms of the series is equal to , where  is a positive integer to be determined.

The first three terms in a geometric series are  ,  ,  , where  is a constant.

Find the value of .

Find the sum of the first 12 terms in this series.

The second and fifth terms of a geometric series are 13.44 and 5.67 respectively.  The series has first term  and common ratio

By first determining the values of  and , calculate the sum to infinity of the series.

Calculate the difference between the sum to infinity of the series and the sum of the first 20 terms of the series. Give your answer accurate to 2 decimal places.

A geometric series has first term 9, and the sum of the first three terms of the series is 19.  The common ratio of the series is .

Show that .

Find the two possible values of .

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

The first three terms of a geometric sequence are given by   , , and  respectively, where  is a non-zero real number.

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

The third term of a geometric sequence is 12, and the nineteenth term of the same sequence is 47.  The common ratio of the sequence is  .

Show that one possible value of  satisfies the equation   .

By solving the equation in (a), find the corresponding value of correct to 3 significant figures.

Find the other possible value of  (also correct to 3 significant figures), and explain why it is not a valid solution to the equation in (a).

A geometric series has second term 648 and fifth term 375.

Find the smallest value of  such that the sum of the first  terms of the series is greater than 4660.

The sum of the first four terms in a geometric series is 27.2, and the sum of the first eight terms in the same series is 164.9.

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

A geometric series has first term , and its terms are connected by the relationship  for all .

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

where  and  are positive integers and  is a surd.

The first three terms in a geometric series are  , , , where  is a constant.

Find the possible values of .

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

The second and third terms of a convergent geometric series are  and , where  is a real number not equal to 1 or -1.

Find the range of possible values of .

Given that the sum to infinity of the series is -6,

find the two possible values of  .

The geometric series  is convergent, and the sum to infinity of the series is .  The first term of the series is , and the common ratio is .

A different series  is formed by squaring all the terms of the series  above.

Show that  is also a convergent geometric series.

The sum to infinity of the series   is .

Express the ratio in terms of  and , simplifying your answer as far as possible.

Show that if , then  for all . 
Comment on what this shows about the relationship between the terms of the two series.