Edexcel International A Level Maths: Pure 2

Topic Questions

International A LevelMaths: Pure 2EdexcelTopic Questions4. Sequences & Series4.4 Sequences & Series

4.4 Sequences & Series

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1a
Sme Calculator2 marks

Calculate

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

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1b
Sme Calculator2 marks

The sum given in part [a] is an arithmetic series.
Write down the first term and the common difference.

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2a
Sme Calculator2 marks

Calculate

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

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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
Sme Calculator3 marks

It is given that

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

where  is a positive integer. 

(i)
Show that 18 a 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
Sme Calculator2 marks

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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4b
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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5
Sme Calculator2 marks

Given that

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

determine the value of k.

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1
Sme Calculator4 marks

Given that sum from r equals 1 to k of 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
Sme Calculator4 marks

 Given that  sum from n equals 1 to 9 of open parentheses a plus open parentheses n minus 1 close parentheses d close parentheses equals negative 279 and  sum from n equals 1 to 13 of 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
Sme Calculator4 marks

Given that  sum from r equals 1 to k of 7 cross times 3 to the power of r 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 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... 

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 open parentheses negative 4 x close parentheses to the power of n minus 1 end exponent space equals space 24    

Calculate the value of x.

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5a
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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5b
Sme Calculator6 marks

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

       sum from r equals 1 to n of open parentheses 2 r close parentheses squared equals 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 open parentheses 2 r minus 1 close parentheses squared equals fraction numerator n 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 u subscript r equals sum from r equals 1 to 100 of r.

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1a
Sme Calculator2 marks

The first  k terms of a series are given by sum from r equals 1 to k of 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 open parentheses 7 plus 5 r close parentheses equals 1190, 

(i)
Show that left parenthesis 5 k plus 119 right parenthesis left parenthesis k minus 20 right parenthesis equals 0
(ii)
Hence find the value of k.

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2a
Sme Calculator2 marks

The first k terms of a series are given by sum from r equals 1 to k of 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
Sme Calculator3 marks

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

Show that k equals 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 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...

Write down the common ratio, r , of the series.

Given that the series is convergent, and that  sum from n equals 1 to infinity of open parentheses 2 x close parentheses to the power of n minus 1 end exponentequals 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...

Given that  sum from n equals 1 to 7 of open parentheses a plus open parentheses n minus 1 close parentheses d close parentheses equals 91 and  sum from n equals 1 to 10 of 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 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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5b
Sme Calculator2 marks

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

               sum from r equals 1 to n of r squared equals fraction numerator n open parentheses n plus 1 close parentheses 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 u subscript r        

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5c
Sme Calculator3 marks

Find the value of  51 squared plus 52 squared plus 53 squared plus...plus space 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 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 3 cross times open parentheses negative 2 close parentheses to the power of r equals negative 262146, 

(i)   show that  fraction numerator k minus 1 over denominator 2 end fraction equals 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 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 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 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 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..., 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 square root of 3 cross times open parentheses square root of 2 x end root close parentheses to the power of n minus 1 end exponent equals 3 square root of 3      

Calculate the value of x.

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5
Sme Calculator5 marks

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

Calculate sum from r equals 11 to 23 of 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 cos open parentheses k pi close parentheses 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 by

               v subscript k equals sin left parenthesis k q straight 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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7
Sme Calculator4 marks

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

         sum from r equals 1 to n of open parentheses 2 r close parentheses squared minus sum from r equals 1 to n of open parentheses 2 r minus 1 close parentheses squared equals sum from r equals 1 to 2 n of r

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