Geometric Sequences & Series (A Level only) (OCR A Level Maths: Pure)

Exam Questions

3 hours33 questions
1
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4 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.....

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2a
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2 marks

Evaluate

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

2b
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2 marks

Evaluate.

         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

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3
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3 marks

Write down a formula for the nth term of each of the following geometric sequences

(i)
3 comma space 12 comma space 48 comma space 192 comma horizontal ellipsis
(ii)
First term: a equals 5
Common ratio: r equals negative 2
(iii)
a equals 16 comma space space r equals 1 half

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4
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3 marks

Find the 5th and 10th terms in 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
(iii)
u subscript n equals 3 to the power of negative n end exponent

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5a
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3 marks

The 3rd and 6th terms of a geometric sequence are 10 and 270 respectively,
Find the first term and the common ratio.

5b
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2 marks

The 12th term of a geometric sequence is 16 times greater than the 8th term.   
Find the possible values of the common ratio.

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6a
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2 marks

Find the sum of the first 12 terms of the geometric series that has first term 5 and common ratio begin inline style begin display style 3 over 2 end style end style, giving your answer to the nearest whole number.

6b
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2 marks

Find the sum to infinity of the geometric series that has first term 4 and common ratio 1 over 8.

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7a
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2 marks

The first term of a geometric sequence is 2.

The 6th term of the sequence is 486.

The sum of the first  terms is 177 146.

Find the common ratio.

7b
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2 marks

Show that 3 to the power of n equals 177 space 147.

7c
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2 marks

Hence find the value of n.

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8a
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2 marks

The first term of a geometric sequence is 6.

The sum to infinity is 8.

Show that the common ratio is 0.25.

8b
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1 mark

Briefly explain why the geometric sequence with first term 6 and common ratio 0.25 has a sum to infinity.

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9a
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1 mark

A geometric series is given by

             k left parenthesis k plus 1 right parenthesis plus k left parenthesis k plus 1 right parenthesis squared plus k left parenthesis k plus 1 right parenthesis cubed plus k left parenthesis k plus 1 right parenthesis to the power of 4 plus midline horizontal ellipsis

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

Write down a formula for the nth term of the series, in terms of k.

9b
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2 marks

Show that the sum to infinity is negative left parenthesis k plus 1 right parenthesis.

9c
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2 marks

The sum to infinity is negative 1 fourth. Find the value of k.

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1a
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2 marks

The first three terms of a geometric sequence are given by x squared, 4 x, and x plus 14 space respectively, where x greater than 0.

Show that x cubed minus 2 x squared equals 0.

1b
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3 marks

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

1c
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1 mark

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

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2a
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3 marks

A geometric sequence has first term 900 and a common ratio r where space r greater than 0.  The 18th term of the sequence is 18.

Show that r satisfies the equation 17 space log r plus log 50 equals 0.

2b
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1 mark

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

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3
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5 marks

A geometric series has first term 19 and common ratio  2 over 3.

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

(i)
show that k greater than fraction numerator log left parenthesis 1 over 57 right parenthesis over denominator log open parentheses 2 over 3 close parentheses end fraction

 

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

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4a
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2 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 series is 11.02.

The common ratio of the series is r.

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

4b
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2 marks

Hence find the two possible values of r.

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5
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4 marks

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

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

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6a
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5 marks

The first three terms in a geometric series 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.
6b
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1 mark

Find the common ratio, r, of this series.

6c
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2 marks

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

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7a
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3 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.

7b
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1 mark

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

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8a
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2 marks

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

Show that the common ratio, r, of the series is 5 over 6.  

8b
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2 marks

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

8c
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2 marks

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

8d
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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.

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1
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5 marks

The first three terms of a geometric sequence are given by x plus 12, 3 x, and x squared respectively, where x is a non-zero real number.

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

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2a
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3 marks

The second term of a geometric sequence is 4, and the sixteenth term of the same sequence is 9.  The common ratio is r, where r greater than 0.

Show that r satisfies the equation, 14 space ln space r plus ln space left parenthesis 4 over 9 right parenthesis space equals 0.

2b
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1 mark

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

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3
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5 marks

A geometric series has first term 14 and common ratio  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.

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4
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4 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 series is 13.23.

Find the common ratio, r, of the series.

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5
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4 marks

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

Show that the sum of the first ten terms of the series is equal to k a left parenthesis square root of 5 plus 1 right parenthesis, where k is a positive integer to be determined.

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6a
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5 marks

The first three terms in a geometric series 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.

6b
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3 marks

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

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7a
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6 marks

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

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

7b
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2 marks

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.

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8a
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3 marks

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 r.

Show that 9 r squared plus 9 r minus 10 equals 0.

8b
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2 marks

Find the two possible values of r.

8c
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3 marks

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

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1
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5 marks

The first three terms of a geometric sequence are given by  x plus 11 , 5 x, and 3 x squared respectively, where  is a non-zero real number.

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

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2a
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3 marks

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  r.

Show that one possible value of r satisfies the equation  16 space ln space r plus ln space left parenthesis 12 over 47 right parenthesis equals 0 .

2b
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1 mark

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

2c
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2 marks

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

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3
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6 marks

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

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

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4
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5 marks

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, r, of the series.

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5
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4 marks

A geometric series has first term a, and its terms are connected by the relationship u subscript n plus 4 end subscript equals 9 u subscript n for all n greater or equal than 1.

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

            S subscript 12 equals k a left parenthesis square root of n plus 1 right parenthesis

where k and n are positive integers and square root of n is a surd.

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6a
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4 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.

6b
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4 marks

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

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7a
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5 marks

The second and third terms of a convergent geometric series are left parenthesis x minus 1 right parenthesis and left parenthesis x squared minus 1 right parenthesis, where x is a real number not equal to 1 or -1.

Find the range of possible values of x.

7b
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4 marks

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

find the two possible values of  x.

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8a
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4 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 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 also a convergent geometric series.

8b
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3 marks

The sum to infinity of the 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.

8c
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6 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.  Comment on what this shows about the relationship between the terms of the two series.

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