A LevelMaths: PureAQAExam Questions4. Sequences and Series4.5 Sequences & Series (A Level only)

Sequences & Series (A Level only) (AQA A Level Maths: Pure)

Exam Questions

3 hours32 questions
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1a
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Calculate

      sum from r equals 1 to 5 of 2 r plus 1

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1b
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The sum given in part (a) is an arithmetic series.
Write down the first term and the common difference.

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2a
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Calculate

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

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2b
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The sum given in part (a) is a geometric series.
Write down the first term and the common ratio.

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3a
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It is given that

      sum from r equals 1 to 4 of a space open parentheses r plus 2 close parentheses equals 72

where a is a positive integer. 

(i)
Show that 18a space equals 72.
(ii)
Find the value of a.

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3b
Sme Calculator1 mark

Determine if the series is arithmetic or geometric, justifying your answer.

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4a
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A Fibonacci sequence can be expressed as the following recurrence relation

         u subscript n plus 2 end subscript equals u subscript n plus 1 end subscript plus u subscript n comma space space space space space space space space space space space space space n greater or equal than 1

Write down the first six terms of the Fibonacci sequence with u subscript 1 equals u subscript 2 equals 1.

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4b
Sme Calculator2 marks

Find

      sum from r equals 1 to 5 of u subscript r

with u subscript 1 equals 2 comma space u subscript 2 equals 4

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5a
Sme Calculator2 marks

A sequence is defined by the recurrence relation  u subscript n plus 1 end subscript equals 2 u subscript n space end subscript comma space space space space space u subscript 1 equals 5 comma space space space space space n greater or equal than 1..

Write down the first five terms of the sequence.

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5b
Sme Calculator1 mark

Determine if the sequence is arithmetic or geometric, justifying your answer.

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5c
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Find

      sum from r equals 1 to 5 of 2 u subscript n

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6a
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The n to the power of t h end exponent term of an arithmetic series is given by u subscript n equals 3 n plus 5.
Write the sum of the series, up to the n to the power of t h end exponent term, in sigma notation.

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6b
Sme Calculator2 marks

The n to the power of t h end exponent term of a geometric series is given by u subscript n equals 5 cross times 2 to the power of n minus 1 end exponent.
Write the sum of the series, up to the n to the power of t h end exponent term, in sigma notation.

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7
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Given that

         sum from r equals 1 to k of r squared equals 55

determine the value of k.

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8a
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A sequence is defined for n greater or equal than 1 by the recurrence relation  u subscript n plus 1 end subscript equals 2 u subscript n minus 2  with u subscript 1 equals 4.

Calculate

         sum from r equals 1 to 6 of u subscript r

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8b
Sme Calculator1 mark

What value of u subscript 1 would make every term of the sequence equal?

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8c
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Find the range of values for u subscript 1that would ensure every term of the sequence is positive?

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1a
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The first k terms of a series are given by sum from r equals 1 to k of space open parentheses 7 plus 5 r close parentheses.

Show that this is an arithmetic series, and determine its first term and common difference.

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1b
Sme Calculator3 marks

Given that   sum from r equals 1 to k of space open parentheses 7 plus 5 r close parentheses equals 1190 ,

(i)
Show that open parentheses 5 k plus 119 close parentheses open parentheses k minus 20 close parentheses equals 0.
(ii)
Hence find the value of k.

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2a
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The first k terms of a series are given by sum from r equals 1 to k of space 5 cross times 2 to the power of r .

Show that this is a geometric series, and determine its first term and common ratio.

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2b
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Given that  sum from r equals 1 to k of space 5 cross times 2 to the power of r = 20470,

Show that k equals space fraction numerator log space 2048 over denominator log space 2 end fraction

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2c
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For this value of k, calculate sum from r equals 1 to k plus 3 of space 5 cross times 2 to the power of r.

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3
Sme Calculator4 marks

A geometric series is given by 1 plus 2 x plus 4 x squared plus horizontal ellipsis

(i)
Write down the common ratio, r, of the series.
(ii)
Given that the series is convergent, and that sum from n equals 1 to infinity of space 2 x to the power of n minus 1 end exponent equals 19, calculate the value of x.

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4
Sme Calculator4 marks

An arithmetic series is given by a plus left parenthesis a plus d right parenthesis plus left parenthesis a plus 2 d right parenthesis plus horizontal ellipsis

Given that sum from n equals 1 to 7 of space open parentheses a plus open parentheses n minus 1 close parentheses space d close parentheses equals 91 and sum from n equals 1 to 10 of space open parentheses a plus open parentheses n minus 1 close parentheses d close parentheses equals 175, find the values of a and d.

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5a
Sme Calculator2 marks

A sequence is defined for k greater or equal than 1 by the recurrence relation  u subscript k plus 1 end subscript equals u subscript k minus 3 comma space space space u subscript 1 equals 23.

Calculate

     sum from n equals 1 to 10 of space u subscript n

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5b
Sme Calculator3 marks

sum from n equals 11 to 15 of space u subscript n

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6a
Sme Calculator2 marks

A sequence is defined for k greater or equal than 1 by the recurrence relation u subscript k plus 1 end subscript equals u subscript k over 3 space comma space space space space u subscript 1 equals 54.

Calculate, giving your answers as exact values 

      sum from n equals 1 to 9 of space u subscript n

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6b
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sum from n equals 10 to infinity of space u subscript n

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7a
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A sequence is defined for k greater or equal than 1 spaceby the recurrence relation u subscript k plus 1 end subscript equals p u subscript k minus 2 comma space space space u subscript 1 equals 2 comma

where p is a constant.

Write down expressions for u subscript 2 and  u subscript 3 in terms of p.

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7b
Sme Calculator4 marks

Given that the sequence is periodic with order 2, and given as well that u subscript 1 not equal to u subscript 2,

Find the value of p.

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7c
Sme Calculator2 marks

For the value of p found in part (b)

Calculate sum from n equals 1 to 1001 of space u subscript n

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8a
Sme Calculator1 mark

The terms of a sequence are defined by u subscript k equals k squared for all k greater or equal than 1.

State, with a reason, whether this sequence is increasing, decreasing, or neither.

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8b
Sme Calculator2 marks

It can be shown that, for all n greater or equal than 1,

         sum from r equals 1 to n of space r squared space equals fraction numerator n space open parentheses n plus 1 close parentheses space open parentheses 2 n plus 1 close parentheses over denominator 6 end fraction

Using that formula,

Calculate sum from r equals 1 to 50 of space u subscript r

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8c
Sme Calculator3 marks

Find the value of  51 squared plus 52 squared plus 53 squared plus horizontal ellipsis plus 99 squared plus 100 squared,i.e. the sum of the squares of all the integers between 51 and 100 inclusive.

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1
Sme Calculator4 marks

Given that sum from r equals 1 to k of space open parentheses 31 minus 6 r close parentheses equals negative 943, 

(i)
Show that left parenthesis 3 k plus 41 right parenthesis left parenthesis k minus 23 right parenthesis equals 0

(ii)
Hence, find the value of k.

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2
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Given that  sum from n equals 1 to 9 of space open parentheses a plus space open parentheses n minus 1 close parentheses d close parentheses equals negative 279 and sum from n equals 1 to 13 of space open parentheses a plus open parentheses n minus 1 close parentheses d close parentheses equals negative 585, find the values of a and d.

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3a
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Given that sum from r equals 1 to k of space 7 cross times 3 to the power of r space space equals 620004,

Show that k equals fraction numerator log space 59049 over denominator log space 3 end fraction

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3b
Sme Calculator3 marks

For this value of k, calculate sum from r equals 0 to k plus 3 of space 7 cross times 3 to the power of r.

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4a
Sme Calculator3 marks

A convergent geometric series is given by 1 minus 4 x plus 16 x squared minus 64 x cubed plus horizontal ellipsis

Write down the range of possible values of x.

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4b
Sme Calculator3 marks

Given that  sum from n equals 1 to infinity of space open parentheses negative 4 x close parentheses to the power of n minus 1 end exponent space=24

Calculate the value of x.

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5a
Sme Calculator3 marks

A sequence is defined for k greater or equal than 1 by the recurrence relation  u subscript k plus 1 end subscript equals u subscript k plus 7 comma space space space space space space u subscript 1 equals 23.

Calculate

sum from n equals 15 to 25 of space u subscript n

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5b
Sme Calculator2 marks

sum from n equals 1 to 25 of space open parentheses u subscript n space minus 3 close parentheses

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6a
Sme Calculator3 marks

A sequence is defined for k greater or equal than 1 spaceby the recurrence relation  u subscript k plus 1 end subscript equals fraction numerator 2 u subscript k over denominator 7 end fraction comma space space space space space space space u subscript 1 equals 686.

Calculate, giving your answers as exact values

stack sum space with n equals 7 below and infinity on top space u subscript n 

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6b
Sme Calculator2 marks

sum from n equals 1 to infinity of space u subscript n plus 4 end subscript

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7a
Sme Calculator5 marks

A sequence is defined for k greater or equal than 1 by the recurrence relation u subscript k plus 1 end subscript equals left parenthesis p minus 2 right parenthesis u subscript k minus 2 comma space space space space space space space space space space u subscript 1 equals 3

where p is a constant.

Given that the sequence is periodic with order 2, and given as well that u subscript 1 not equal to u subscript 2,

Find the value of  p.

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7b
Sme Calculator2 marks

For the value of p found in part (a),

Calculate  sum from n equals 50 to 900 of space u subscript n

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8a
Sme Calculator1 mark

The terms of a sequence are defined, for all k greater or equal than 1 comma  by  u subscript k equals left parenthesis negative 1 right parenthesis to the power of k cross times k squared.

State, with a reason, whether this sequence is increasing, decreasing, or neither.

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8b
Sme Calculator6 marks

It can be shown that, for all n greater or equal than 1,

            sum from r equals 1 to n of space open parentheses 2 r close parentheses squared= fraction numerator 2 n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses over denominator 3 end fraction    and     sum from r equals 1 to n of space open parentheses 2 r minus 1 close parentheses squared equals fraction numerator n space open parentheses 2 n plus 1 close parentheses open parentheses 2 n minus 1 close parentheses over denominator 3 end fraction        

Using those formulas,

Show that sum from r equals 1 to 100 of space u subscript r space equals space sum from r equals 1 to 100 of space r.

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1
Sme Calculator4 marks

Given that  sum from r equals 1 to k of open parentheses 89 minus 5 r close parentheses equals negative 35, find the value of k.

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2a
Sme Calculator5 marks

Given that sum from r equals 1 to k of space 3 cross times open parentheses negative 2 close parentheses to the power of r space equals negative 262146,  

(i)
show that fraction numerator k minus 1 over denominator 2 end fraction equals space fraction numerator log space 65536 over denominator log space 4 end fraction    
(ii)
hence find the value of k.

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2b
Sme Calculator3 marks

For this value of k, calculate sum from r equals 5 to k plus 2 of space 3 cross times open parentheses negative 2 close parentheses to the power of r.

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3
Sme Calculator5 marks

Given that sum from n equals 7 to 12 of space open parentheses a plus open parentheses n minus 1 close parentheses d close parentheses equals negative 69,  sum from n equals 7 to 16 of space open parentheses a plus open parentheses n minus 1 close parentheses d close parentheses equals negative 175 and  sum from n equals 1 to 6 of space open parentheses a plus open parentheses n minus 1 close parentheses d close parentheses equals negative 13 d, find the values of a and d.

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4a
Sme Calculator4 marks

A convergent geometric series is given by square root of 3 plus square root of 6 x end root plus 2 x square root of 3 plus horizontal ellipsis space comma space, where in all cases the square root symbol indicates the positive square root of the number in question.

Write down the range of possible values of x.

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4b
Sme Calculator3 marks

Given that sum from n equals 2 to infinity of space square root of 3 space cross times space open parentheses square root of 2 x end root close parentheses to the power of n minus 1 end exponent space equals 3 square root of 3

Calculate the value of  x.

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5
Sme Calculator5 marks

A sequence is defined for space k greater or equal than 1 by  u subscript k equals square root of 13 plus left parenthesis negative 2 right parenthesis to the power of k minus 1 end exponent.

Calculate sum from r equals 11 to 23 of space u subscript r, giving your answer as an exact value.

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6a
Sme Calculator3 marks

A sequence is defined for all k greater or equal than 1 by

         u subscript k equals negative 2 k cross times left parenthesis c o s left parenthesis k pi right parenthesis right parenthesis to the power of k plus 1 end exponent

Determine, giving reasons for your answer, whether the sequence is increasing, decreasing, or neither.

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6b
Sme Calculator2 marks

A different sequence is defined for all k greater or equal than 1 spaceby v subscript k equals s i n left parenthesis k q pi right parenthesis

where q is a real constant.

Given that the sequence is not periodic,

suggest a possible value for q, giving a reason for your answer.

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7a
Sme Calculator3 marks

A sequence is defined for k greater or equal than 1 by the recurrence relation  u subscript k plus 2 end subscript equals u subscript k plus 1 end subscript over u subscript k comma space space space space space space u subscript 1 equals a comma space space space space space space space u subscript 2 equals b

where a and b are real numbers.

Show that the sequence is periodic, and determine its order.

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7b
Sme Calculator5 marks

Given that sum from r equals 1 to 44 of space u subscript r space equals negative 50 and   sum from r equals 1 to 84 of space u subscript r space equals negative 92

determine the possible values of a and b.

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8
Sme Calculator4 marks

Prove that, for all n greater or equal than 1,

         sum from r equals 1 to n of space open parentheses 2 r close parentheses to the power of 2 space space end exponent minus sum from r equals 1 to n of space open parentheses 2 r minus 1 close parentheses squared space equals sum from r equals 1 to 2 n of space r

 

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