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

Exam code: 9MA0

3 hours37 questions
14 marks

Identify which of the following are geometric sequences.

For those that are, write down the first term and the common ratio.

(i) 3 comma 8 comma 13 comma 18 comma horizontal ellipsis

(ii) 5 comma 15 comma space 45 comma 135 comma horizontal ellipsis

(iii) 5 comma negative 10 comma space 20 comma negative 40 comma horizontal ellipsis

(iv) 1 third comma space 1 over 6 comma space 1 over 12 comma space.....

Did this page help you?

23 marks

A geometric sequence has a first term, a, a common ratio, r, and an nth term, u subscript n, where n element of straight natural numbers.

Find a formula for u subscript n for each of the following geometric sequences:

(i) 3 comma space 12 comma space 48 comma space 192 comma space horizontal ellipsis

(ii) a equals 5r equals negative 2

(iii) a equals 16, r equals 1 half

Did this page help you?

3
Sme Calculator
3 marks

Find the 5th and 10th terms of each of the following geometric sequences:

(i) u subscript n equals 2 left parenthesis 3 right parenthesis to the power of n

(ii) u subscript n equals 10 space 000 left parenthesis 1.02 right parenthesis to the power of n giving your answers to 2 decimal places

(iii) u subscript n equals 3 to the power of negative n end exponent

Did this page help you?

4a
Sme Calculator
2 marks

A geometric series has a first term of 5 and a common ratio of begin inline style begin display style 3 over 2 end style end style.

Use the formula S subscript n equals fraction numerator a open parentheses 1 minus r to the power of n close parentheses over denominator 1 minus r end fraction to find the sum of the first 12 terms, to the nearest whole number.

4b1 mark

A different geometric series has a first term of 4 and a common ratio of 1 over 8.

Use the formula S subscript infinity equals fraction numerator a over denominator 1 minus r end fraction to find the sum to infinity.

Did this page help you?

5a2 marks

The first term of a geometric series is 6.

The sum to infinity of the series is 8.

Show that the common ratio is 0.25.

5b1 mark

Explain why the sum to infinity exists for the geometric series in part (a).

Did this page help you?

6a1 mark

For a geometric sequence,

  • the first term is 900

  • the common ratio is r where space r greater than 0

  • the 18th term is 18

Show that r satisfies the equation

r to the power of 17 equals 1 over 50

6b
Sme Calculator
1 mark

Find the value of r correct to 3 significant figures.

Did this page help you?

1a
Sme Calculator
1 mark

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.

1b
Sme Calculator
3 marks

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

1c
Sme Calculator
2 marks

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

Did this page help you?

2
Sme Calculator
3 marks

A car has six forward gears.

The fastest speed of the car

  • in 1st gear is 28 km h-1

  • in 6th 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 5th gear.

Did this page help you?

3a2 marks

A geometric sequence is defined as follows

  • The first term is 2

  • The sixth term is 486

Find the common ratio.

3b2 marks

The sum of the first n terms is 177146 .

Show that 

3 to the power of n equals 177147

3c
Sme Calculator
1 mark

Hence find the value of n.

Did this page help you?

4a3 marks

In a geometric sequence

  • the 3rd term is 10

  • the 6th term is 270

Find the first term and the common ratio.

4b2 marks

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

Find the possible values of the common ratio.

Did this page help you?

5a3 marks

The first three terms of a geometric sequence are given by

x squared comma space space 4 x comma space space open parentheses x plus 14 close parentheses

where x greater than 0.

Show that

x cubed minus 2 x squared equals 0

5b
Sme Calculator
3 marks

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

5c
Sme Calculator
1 mark

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

Did this page help you?

6
Sme Calculator
4 marks

In a geometric series,

  • the first term is 19

  • the common ratio is 2 over 3

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

(i) Show that 

k greater than fraction numerator log open parentheses 1 over 57 close parentheses over denominator log open parentheses 2 over 3 close parentheses end fraction

 

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

Did this page help you?

7a
Sme Calculator
3 marks

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 not equal to 1

 Show that

fraction numerator 1 minus r to the power of 4 over denominator 1 minus r squared end fraction equals 58 over 49

7b2 marks

Hence find the possible values of r.

Did this page help you?

8a4 marks

The first three terms in a geometric sequence are left parenthesis k minus 3 right parenthesis, k, left parenthesis 2 k plus 8 right parenthesis, where k greater than 0 is a constant.

(i) Show that k squared plus 2 k minus 24 equals 0

(ii) Hence find the value of k.

8b1 mark

Find the common ratio of the sequence.

8c
Sme Calculator
2 marks

 Find the sum of the first 12 terms.

Did this page help you?

9a2 marks

A geometric series is defined as follows:

  • The first term is 64

  • The sum to infinity is 384

Show that the common ratio is 5 over 6.  

9b
Sme Calculator
2 marks

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

9c
Sme Calculator
2 marks

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

9d
Sme Calculator
4 marks

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

Did this page help you?

10a2 marks

Given that the geometric series 

negative 1 plus 3 x minus 9 x squared plus 27 x cubed plus horizontal ellipsis 

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

10b1 mark

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

Did this page help you?

113 marks

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 a over b, giving your answer as a fraction in simplest form.

Did this page help you?

12a2 marks

A geometric sequence is given by

k open parentheses k plus 1 close parentheses comma space space k open parentheses k plus 1 close parentheses squared comma space space k open parentheses k plus 1 close parentheses cubed comma space space k open parentheses k plus 1 close parentheses to the power of 4 comma space space...

where k is a constant such that vertical line k plus 1 vertical line less than 1.

Find, in terms of k and n, a formula for the nth term of the sequence.

12b2 marks

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

Show that the sum to infinity of the series is

negative left parenthesis k plus 1 right parenthesis

12c2 marks

Given that the sum to infinity is negative 1 fourth, find the value of k.

Did this page help you?

1a4 marks

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 not equal to 1 and a not equal to 0, prove that

S subscript n equals fraction numerator a open parentheses 1 minus r to the power of n close parentheses over denominator 1 minus r end fraction

1b
Sme Calculator
4 marks

Given also that S subscript 10 is four times S subscript 5, find the exact value of r.

Did this page help you?

2a2 marks

The first three terms of a geometric sequence are

3 k plus 4 space space space space space space space space space space 12 – 3 k space space space space space space space space space space k plus 16

where k is a constant.

Show that k satisfies the equation

3 k squared – 62 k plus 40 equals 0

2b
Sme Calculator
5 marks

Given that the sequence converges,

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

(ii) find the value of S subscript infinity

Did this page help you?

35 marks

The first three terms of a geometric sequence are given by

open parentheses x plus 12 close parentheses comma space space 3 x comma space space x squared

where x is a non-zero real number.

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

Did this page help you?

4a2 marks

Find the value of

sum from r equals 1 to 5 of 3 open parentheses 2 to the power of r close parentheses

4b
Sme Calculator
2 marks

Find the value of

         sum from r equals 4 to 8 of open parentheses negative 1 close parentheses to the power of r open parentheses 2 to the power of r close parentheses

Did this page help you?

5a4 marks

In a geometric sequence,

  • the second term is 4

  • the common ratio is r where space r greater than 0

  • the 16th term is 9

Show that r satisfies the equation

14 ln space r plus ln open parentheses 4 over 9 close parentheses equals 0

5b
Sme Calculator
2 marks

Find the value of r, to 3 significant figures.

Did this page help you?

6
Sme Calculator
5 marks

A geometric series has first term of 14 and common ratio of  99 over 100.

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

Did this page help you?

7
Sme Calculator
5 marks

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.

Did this page help you?

8
Sme Calculator
4 marks

A geometric series has first term of a and common ratio of square root of 5.

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

k a left parenthesis square root of 5 plus 1 right parenthesis

where k is a positive integer to be found.

Did this page help you?

9a3 marks

The first three terms in a geometric sequence are left parenthesis 2 k plus 3 right parenthesiskleft parenthesis k minus 2 right parenthesis, where k less than 0 is a constant.

Find the value of k.

9b
Sme Calculator
3 marks

Find the sum of the first 12 terms.

Did this page help you?

10a
Sme Calculator
6 marks

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.

10b
Sme Calculator
3 marks

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.

Did this page help you?

11a4 marks

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 not equal to 1

Show that

9 r squared plus 9 r minus 10 equals 0

11b2 marks

Find the possible values of r.

11c3 marks

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

Did this page help you?

1a3 marks

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 space space space space space space space space space space space 5 plus 2 sin theta space space space space space space space space space space spaceandspace space space space space space space space space space space 6 tan theta

show that

4 sin squared theta space minus space 52 sin theta plus 25 equals 0

1b2 marks

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

1c5 marks

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

k open parentheses 1 minus square root of 3 close parentheses

where k is a constant to be found.

Did this page help you?

2
Sme Calculator
5 marks

The first three terms of a geometric sequence are given by

open parentheses x plus 11 close parentheses comma space space 5 x comma space space 3 x squared

 where x is a non-zero real number.

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

Did this page help you?

3
Sme Calculator
6 marks

In a geometric series,

  • the second term is 648

  • the fifth term is 375

  • the sum of the first kterms of the series is greater than 4660

Find the smallest value of k.

Did this page help you?

4
Sme Calculator
5 marks

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.

Did this page help you?

5
Sme Calculator
5 marks

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

u subscript n plus 4 end subscript equals 9 u subscript n

for all n element of straight natural numbers.

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

k a left parenthesis square root of n plus 1 right parenthesis

where k and n are positive integers to be found.

Did this page help you?

6a4 marks

The first three terms in a geometric series are left parenthesis 2 k plus 6 right parenthesis, k, left parenthesis k minus 4 right parenthesis, where k is a constant.

Find the possible values of k.

6b4 marks

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

Did this page help you?

7a3 marks

In a convergent geometric series

  • the second term is open parentheses x minus 1 close parentheses

  • the third term is open parentheses x squared minus 1 close parentheses

where x squared not equal to 1.

Find the range of possible values of x.

7b5 marks

The sum to infinity of the series is negative 6

Find the possible values of  x.

Did this page help you?

8a3 marks

The geometric series S equals u subscript 1 plus u subscript 2 plus u subscript 3 plus horizontal ellipsis plus u subscript n plus horizontal ellipsis is convergent, and the sum to infinity of the series is S subscript infinity.  The first term of the series is a, and the common ratio is r.

A different series T equals u subscript 1 superscript 2 plus u subscript 2 superscript 2 plus u subscript 3 superscript 2 plus horizontal ellipsis plus u subscript n superscript 2 plus horizontal ellipsis is formed by squaring all the terms of the series S above.

Show that T is also a convergent geometric series.

8b
Sme Calculator
3 marks

The sum to infinity of series T is T subscript infinity.

Express the ratio T subscript infinity over S subscript infinity spacein terms of a and r, simplifying your answer as far as possible.

8c6 marks

Show that if T subscript infinity equals S subscript infinity, then 

u subscript k superscript 2 equals u subscript 2 k minus 1 end subscript plus u subscript 2 k end subscript

for all k greater or equal than 1

Hence describe the relationship between the terms of the two series in the case when T subscript infinity equals S subscript infinity.

Did this page help you?