Arithmetic Progressions (CIE A Level Maths: Pure 1)

Exam Questions

3 hours33 questions
1
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2 marks

Write down the next three terms in these arithmetic sequences

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

The first three terms of an arithmetic progression are k , 2 k  and 3 k,  where k is a constant.

Write down a formula for the n to the power of t h end exponentterm of the progression, 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 open parentheses n plus space 1 close parentheses.

8c
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2 marks

The sum of the first 12 terms of the progression 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  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 first three terms of an arithmetic progression are k plus 7 comma space space2 k plus space 18  and 3 k space plus space 29. The nth term is 36.

      Show that n space equals space fraction numerator 40 over denominator k plus space 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 space plus space 860 over denominator k space plus space 11 end fraction.

6c
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1 mark

Given that the sum of the first n terms is 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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2 marks

The th term of an arithmetic progression is given byu subscript k equals 26 minus 3 k.

Calculate

The sum of the first ten terms of the progression.

8b
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3 marks

The sum of the eleventh through fifteenth terms of the progression.

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9a
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3 marks

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

      1 space plus space 3 space plus space 5 space plus... space plus space 149

9b
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4 marks

Below is an arithmetic progression

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

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

In terms of k, find an expression for the number of terms in the progression.

Show that the sum of the terms in the progression is 180 space plus fraction numerator 64800 over denominator k space end fraction.

9c
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2 marks

The first three terms of an arithmetic sequence are 3 q minus 7 comma space5 q space minus space 4 and 7 q space minus space 1.  In terms of q , find the 100th term of the sequence.  Give your answer in simplest form.

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1
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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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2
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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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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

The first three terms of an arithmetic progression are 7 k,14 k and 21 k,  where k is an integer.  The nth term is 1008.

In terms ofspace k, find an expression for the number of terms in this progression.

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

Show that the sum of the first  terms of the progression is 504 space plus space space 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

The first three terms in an arithmetic progression are k space plus space 13 comma 2 k space plus space 9  and 3 k space plus space 5.  The nth term of the progression is denoted by u subscript n.

Find an expression for 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 the sum of the first n terms of the progression is negative space 11.

Find the value of k.

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9a
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3 marks

The kth term of an arithmetic progression is given by u subscript k space equals space 16 space plus space 7 k.

Calculate

The sum of the fifteenth through twenty-fifth terms of the progression.

9b
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2 marks

The sum of the first 25 terms of the progression whose kth term is given by , v subscript k equals u subscript k minus 3 comma where u subscript kis defined as above.

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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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4 marks

The kth term of an arithmetic progression is given by u subscript k space equals space 89 space minus space 5 k. Given that the sum of the first n terms is equal to negative 35,  find the value of n.

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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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5 marks

An arithmetic progression has first term  and common difference d

Given that

the sum of the seventh through twelfth terms is negative 69,

the sum of the seventh through sixteenth terms is negative 175,  and

the sum of the first six terms is negative space 13 d,

find the values of a and d.

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