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

Exam Questions

3 hours32 questions
1
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2 marks

Write down the next three terms in these arithmetic sequences

(i)
30, 18,  6,  ...
(ii)
1 fourth comma space 5 over 12 comma 7 over 12 comma 3 over 4 comma.....

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2
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2 marks

Find the sum of the first four terms in the sequence defined by  u subscript n equals 2 n plus 3.
Justify why this sequence is an arithmetic sequence.

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3
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3 marks

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

(i)
16, 20,  24,  ...
(ii)
First term: a equals 3
Common difference: d equals negative 3
(iii)
a equals 2 comma space space d equals 6

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4
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3 marks

Find the 10th and 20th terms in each of the following arithmetic sequences

(i)
u subscript n equals 4 plus 5 n
(ii)
u subscript n equals begin inline style 1 half end style minus begin inline style 1 fourth end style n
(iii)
u subscript n equals 50 minus 5 n

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5a
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3 marks

The 4th and 8th terms of an arithmetic sequence are 20 and 64 respectively.
Find the first term and the common difference.

5b
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2 marks

The 12th and 16th terms of an arithmetic sequence differ by 20.
Find the possible values of the common difference.

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6
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2 marks

Find the sum of the first 20 terms of the arithmetic series that has first term 3 and common difference 4.

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7a
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2 marks

The first term of an arithmetic sequence is 3.
The 10th term of the sequence is 30.
The sum of the first  terms is 630.

 Find the common difference.

7b
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2 marks

Show that  n squared plus n minus 420 equals 0..

7c
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2 marks

Hence find the value of n.

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8a
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1 mark

An arithmetic series is given by

                       k plus 2 k plus 3 k plus 4 k plus midline horizontal ellipsis

where k is a constant.

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

8b
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2 marks

Show that the sum of the first n terms is fraction numerator k n over denominator 2 end fraction space left parenthesis n plus 1 right parenthesis.

8c
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2 marks

The sum of the first 12 terms is 39.
Find the value of k.

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1
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3 marks

The first three terms in an arithmetic sequence are left parenthesis p minus 9 right parenthesis comma negative 7 comma left parenthesis 9 minus 3 p right parenthesis comma horizontal ellipsis

Find the value of p.

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2
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3 marks

The first three terms in an arithmetic sequence are negative 1 comma space left parenthesis q plus 1 right parenthesis comma space q squared comma horizontal ellipsis

Find the possible values of q.

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3
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4 marks

An arithmetic sequence has first term r squared and common difference 3 r, where r greater than 0.  The fifth term of the sequence is 85.

Find:

(i)
the value of r

(ii)
the ninth term in the sequence.

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4a
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4 marks

The third term of an arithmetic series is 2.  The twelfth term is 65.  The sum of the first n terms is 390.

Show that  7 n squared minus 31 n minus 780 equals 0..

4b
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1 mark

Hence find the value of n.

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5
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3 marks

The sum of the first ten terms in an arithmetic series is 40.  The sum of the first twenty terms in the same series is 280.  Find the first term, a, and the common difference,d , of the series.

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6a
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2 marks

The sum of the first n terms of an arithmetic series is

         S subscript n equals left parenthesis k plus 7 right parenthesis plus left parenthesis 2 k plus 18 right parenthesis plus left parenthesis 3 k plus 29 right parenthesis plus horizontal ellipsis plus 36

Show that  n equals fraction numerator 40 over denominator k plus 11 end fraction.

6b
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3 marks

Hence show that the sum of the first n terms is  fraction numerator 20 k plus 860 over denominator k plus 11 end fraction.

6c
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1 mark

Given that S subscript n equals 180, find the value of k.

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7a
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3 marks

The fifth term of an arithmetic series is k, where k is a constant, and the sum of the first eight terms of the series is 2 k.

Show that the first term, a, of the series is negative 5 k.

7b
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2 marks

Find an expression for the common difference, d, of the series in terms of k.

7c
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2 marks

Given that the ninth term of the sequence is 14, calculate:

the value of k

7d
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2 marks

The sum of the first 30 terms of the series.

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8a
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3 marks

Calculate the sum of all the odd numbers between 0 and 150,

        1 plus 3 plus 5 plus space horizontal ellipsis plus 149

8b
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4 marks

An arithmetic series is defined by

        k plus 2 k plus 3 k plus space horizontal ellipsis plus 360

where k is an integer and a positive factor of 360.

(i)
In terms of k, find an expression for the number of terms in this series.

(ii)
Show that the sum of this series is  180 plus 64800 over k.

8c
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2 marks

In terms of q, find the 100th term of the arithmetic sequence defined by

         space left parenthesis 3 q minus 7 right parenthesis comma space left parenthesis 5 q minus 4 right parenthesis comma space left parenthesis 7 q minus 1 right parenthesis comma horizontal ellipsis space

Give your answer in simplest form.

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1
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3 marks

The first three terms in an arithmetic sequence are left parenthesis q minus 2 right parenthesis comma space q squared comma space left parenthesis 4 q plus 5 right parenthesis comma horizontal ellipsis

Find the possible values of q.

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2
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3 marks

The first two terms in an arithmetic sequence are left parenthesis p plus 15 right parenthesis space and 3.  The fourth term is left parenthesis 3 p minus 16 right parenthesis.

Find the value of p.

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3
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4 marks

An arithmetic sequence has first term r squared and common difference 2 r, where r greater than 0.  The fourth term in the sequence is 4.

Find the value of r, giving your answer as an exact value.

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4
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5 marks

The third term of an arithmetic series is 32.  The eleventh term is 0.  The sum of the first n terms is -44.

Find the value of n.

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5
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4 marks

The sum of the first twelve terms in an arithmetic series is 654.  The sum of the first twenty terms in the same series is 530.  Find the 21st term.

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6a
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3 marks

Prove that the sum of the first n odd numbers is a square number for any value of n greater or equal than 1.

6b
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5 marks

An arithmetic series is defined by

         7 k plus 14 k plus 21 k plus space horizontal ellipsis plus 1008

where k is an integer.

(i)
In terms of k, find an expression for the number of terms in this series.

(ii)
In addition to being an integer, what other two conditions must k satisfy for this to be a valid arithmetic series?

(iii)
Show that the sum of this series is  504 plus 72576 over k.

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7a
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5 marks

The seventh term of an arithmetic series is 3 k, where k is a constant, and the sum of the first nine terms of the series is 4 k minus 3.

In terms of k, find expressions for the first term and common difference of the series.

7b
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4 marks

Given that the nineteenth term of the sequence is 57, find the sum of the first 25 terms of the series.

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8a
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2 marks

An arithmetic series is defined by

          S subscript n equals left parenthesis k plus 13 right parenthesis plus left parenthesis 2 k plus 9 right parenthesis plus left parenthesis 3 k plus 5 right parenthesis plus horizontal ellipsis plus u subscript n plus horizontal ellipsis

Find an expression for n in terms of u subscript n and k.

8b
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5 marks

For a particular value of nu subscript n equals negative 16  and S subscript n equals negative 11.

Find the value of k.

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1
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4 marks

The first four terms in an arithmetic sequence are left parenthesis 2 q minus p right parenthesis comma space 1 comma space left parenthesis p minus q right parenthesis comma space left parenthesis 3 p minus 4 q right parenthesis comma horizontal ellipsis

Find the values of p and q.

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2
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4 marks

The first three terms in an arithmetic sequence are 2 comma space 3 q squared comma space q comma horizontal ellipsis

Given that the first three terms in the sequence are all positive, find the fortieth term in the sequence.

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3
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6 marks

An arithmetic sequence has first term r squared and common difference r, where r greater than 0.  The ninth term in the sequence is 7.

S spaceis the set of all numbers that are in the sequence.  T is the set of all rational numbers.

Determine the elements of the set S intersection T.

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4
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3 marks

The kth term of an arithmetic series is 0.  The sum of the first n terms is also 0.

Find the value of n in terms of  k, giving clear reasons for your answer.

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5
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6 marks

The sum of the first 20 terms in an arithmetic series is 290.  The sum of the first 24 terms in the same series is -180.  In general, the sum of the first  n terms is S subscript n.  Find the greatest value attained by S subscript n for any n greater or equal than 1.

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6
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4 marks

The sum of the first 24 terms in an arithmetic series is nine times the sum of the first two terms in the series.

Find the sum of the first 90 terms in the series.

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7
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7 marks

The nth term in an arithmetic sequence can be expressed in the form

      u subscript n equals left parenthesis n plus 1 half right parenthesis spaceIn 4

Show that the sum of the first  terms of the sequence is

     S subscript n equals In 2straight f open parentheses n close parentheses

where straight f open parentheses n close parentheses is a polynomial in n to be found.

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8a
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4 marks

An arithmetic sequence is defined by

            left parenthesis negative 21 plus ln space 3 space right parenthesis comma space left parenthesis negative 20 plus ln space 27 space right parenthesis comma space left parenthesis negative 19 plus ln space 243 space right parenthesis comma horizontal ellipsis

Find, in terms of n and  ln space 3, an expression for the sum of the first n terms of the sequence.

8b
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2 marks

Hence find the smallest value of n for which the sum of the first  n terms of the sequence is greater than zero.

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