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General Sequences & Series (Edexcel A Level Maths: Pure): Exam Questions

Exam code: 9MA0

3 hours35 questions
Easy
Medium
Hard
Very Hard
1a2 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by the recurrence relation 

table row cell u subscript 1 end cell equals 5 row cell u subscript n plus 1 end subscript end cell equals cell 2 u subscript n end cell end table.

Find the first five terms of the sequence.

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1b2 marks

Hence find the value of

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

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22 marks

By summing each term, find the value of

sum from r equals 1 to 5 of open parentheses 2 r plus 1 close parentheses

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3a1 mark

The n to the power of t h end exponent term of sequence is given by u subscript n equals 3 n plus 5.

Write down an expression for the sum of the first n terms using sigma notation.

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3b1 mark

The n to the power of t h end exponent term of a different sequence is given by u subscript n equals 5 cross times 2 to the power of n minus 1 end exponent.

Write down an expression for the sum of the first n terms using sigma notation.

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42 marks

By summing each term, find the value of

sum from r equals 1 to 3 of 2 cross times 3 to the power of r

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5a2 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by the recurrence relation

table row cell u subscript 1 end cell equals 4 row cell u subscript n plus 1 end subscript end cell equals cell 2 u subscript n minus 2 end cell end table

By finding the first six terms, calculate the value of

         sum from r equals 1 to 6 of u subscript r

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5b1 mark

Describe what happens to the sequence when the first term is changed to u subscript 1 equals 2.

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5c1 mark

Find the range of values of u subscript 1for which every term in the sequence is positive.

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62 marks

Given that

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

find the value of k.

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7a3 marks

It is given that

      sum from r equals 1 to 4 of a 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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7b1 mark

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

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1a2 marks

A sequence of terms a subscript 1 comma space a subscript 2 comma space a subscript 3 comma space... is defined by

a subscript 1 equals 3

a subscript n plus 1 end subscript equals 8 minus a subscript n

(i) Show that this sequence is periodic.

(ii) State the order of this periodic sequence.

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1b
Sme Calculator2 marks

Find the value of

sum from n equals 1 to 85 of a subscript n

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2
Sme Calculator3 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by

u subscript n plus 1 end subscript equals 1 over u subscript n comma space space space space space space u subscript 1 equals 2 over 3

Find the exact value of sum from r equals 1 to 100 of u subscript r.

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3a3 marks

The sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by

u subscript n plus 1 end subscript equals k minus 24 over u subscript n space space space space space u subscript 1 equals 2

where k is an integer.

Given that u subscript 1 plus 2 u subscript 2 plus u subscript 3 equals 0, show that

3 k squared minus 58 k plus 240 equals 0

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3b
Sme Calculator2 marks

Find the value of k, giving a reason for your answer.

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3c1 mark

Find the value of u subscript 3.

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4a3 marks

A sequence of numbers a subscript 1 comma space a subscript 2 comma space a subscript 3 comma space... is defined by

a subscript n plus 1 end subscript equals fraction numerator k open parentheses a subscript n plus 2 close parentheses over denominator a subscript n end fraction space space space space space space space space n element of straight natural numbers

where k is a constant.

Given that

  • the sequence is a periodic sequence of order 3

  • a subscript 1 equals 2

show that

k squared plus k minus 2 equals 0

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4b1 mark

For this sequence explain why k not equal to 1.

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4c
Sme Calculator3 marks

Find the value of

sum from r equals 1 to 80 of a subscript r

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5a2 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by

table row cell u subscript 1 end cell equals 23 row cell u subscript n plus 1 end subscript end cell equals cell u subscript n minus 3 end cell end table

Find the value of

     sum from n equals 1 to 10 of u subscript n

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5b3 marks

Find the value of

sum from n equals 11 to 15 of u subscript n

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6a
Sme Calculator2 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by

table row cell u subscript 1 end cell equals 54 row cell u subscript n plus 1 end subscript end cell equals cell u subscript n over 3 end cell end table

Find the exact value of

      sum from n equals 1 to 9 of u subscript n

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6b
Sme Calculator3 marks

Find the exact value of

sum from n equals 10 to infinity of u subscript n

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7a1 mark

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by

u subscript k equals k squared

 for all k element of straight natural numbers.

State whether this sequence is increasing, decreasing, or neither.

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7b
Sme Calculator2 marks

A formula is given by

         sum from k equals 1 to n of space k 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

Use the formula to find the value of

sum from k equals 1 to 50 of u subscript k

where u subscript k is the sequence in part (a).

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7c
Sme Calculator3 marks

Find the sum of the squares of all the integers between 51 and 100 inclusive,

51 squared plus 52 squared plus 53 squared plus horizontal ellipsis plus 99 squared plus 100 squared

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8a2 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by

table row cell u subscript 1 end cell equals 2 row cell u subscript n plus 1 end subscript end cell equals cell p u subscript n minus 2 end cell end table

where p is a constant.

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

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8b4 marks

The sequence is periodic with order 2.

Find the value of p.

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8c2 marks

Using the value of p in part (b), find

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

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9a2 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by u subscript n equals 7 plus 5 n where n element of straight natural numbers.

Describe this sequence.

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9b3 marks

Given that sum from n equals 1 to k of u subscript n 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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10a2 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by u subscript n equals 5 cross times 2 to the power of n where n element of straight natural numbers.

Describe this sequence.

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10b3 marks

Given that 

sum from n equals 1 to k of space 5 cross times 2 to the power of n equals 20470

find the value of k.

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10c
Sme Calculator2 marks

Use the value of k in part (b) to find

sum from n equals 1 to k plus 3 of 5 cross times 2 to the power of n

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1a
Sme Calculator3 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 space... is defined by

table row cell u subscript 1 end cell equals 35 row cell u subscript n plus 1 end subscript end cell equals cell u subscript n plus 7 cos open parentheses fraction numerator n pi over denominator 2 end fraction close parentheses minus 5 open parentheses negative 1 close parentheses to the power of n end cell end table

(i) Show that u subscript 2 equals 40

(ii) Find the value of u subscript 3 and the value of u subscript 4

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1b
Sme Calculator3 marks

Given that the sequence is periodic with order 4

(i) write down the value of u subscript 5

(ii) find the value of sum from r equals 1 to 25 of u subscript r

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2
Sme Calculator3 marks

Show that

sum from n equals 2 to infinity of open parentheses 3 over 4 close parentheses to the power of n cos open parentheses 180 n close parentheses degree equals 9 over 28

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3
Sme Calculator3 marks

Find the value of

sum from r equals 4 to infinity of 20 cross times open parentheses 1 half close parentheses to the power of r

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44 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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5a4 marks

Given that

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

Find the value of k.

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5b
Sme Calculator3 marks

Use the value of k in part (a) to find

sum from r equals 0 to k plus 3 of 7 cross times 3 to the power of r

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6a
Sme Calculator3 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by

table row cell u subscript 1 end cell equals 23 row cell u subscript n plus 1 end subscript end cell equals cell u subscript n plus 7 end cell end table

Find the value of

sum from n equals 15 to 25 of u subscript n

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6b
Sme Calculator2 marks

Find the value of

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

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7a
Sme Calculator4 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by

table row cell u subscript 1 end cell equals 686 row cell u subscript n plus 1 end subscript end cell equals cell fraction numerator 2 u subscript n over denominator 7 end fraction end cell end table

Find the exact value of

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

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7b
Sme Calculator3 marks

Find the value of

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

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84 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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9a
Sme Calculator4 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 space equals negative 262146

find the value of k.

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9b
Sme Calculator3 marks

Use the value of k in part (a) to find

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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10a5 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by

table row cell u subscript 1 end cell equals 3 row cell u subscript n plus 1 end subscript end cell equals cell open parentheses p minus 2 close parentheses u subscript n minus 2 end cell end table

where p is a constant.

Given that the sequence is periodic with order 2, find the value of p.

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10b3 marks

Use the value of p in part (a) to find

sum from n equals 50 to 900 of u subscript n

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13 marks

Show that

sum from n equals 1 to 48 of log subscript 5 open parentheses fraction numerator n plus 2 over denominator n plus 1 end fraction close parentheses equals 2

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2
Sme Calculator4 marks

Show that sum from r equals 1 to 16 of open parentheses 3 plus 5 r plus 2 to the power of r close parentheses equals 131798

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35 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

find the value of x.

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46 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

  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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5
Sme Calculator5 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by

u subscript n equals square root of 13 plus left parenthesis negative 2 right parenthesis to the power of n minus 1 end exponent

where n element of straight natural numbers.

Find the exact value of

sum from n equals 11 to 23 of u subscript n

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6a2 marks

A sequence u subscript 1 comma space u subscript 2 comma space u subscript 3 comma space... is defined by

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

where k element of straight natural numbers.

Determine, with reason, whether the sequence is increasing, decreasing, or neither.

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6b2 marks

A different sequence v subscript 1 comma space v subscript 2 comma space v subscript 3 comma space... is defined by 

v subscript k equals sin left parenthesis k q pi right parenthesis

where k element of straight natural numbers and q is a rational constant.

Explain why v subscript k must be periodic.

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7a4 marks

A sequence is defined by the recurrence relation  

u subscript 1 equals a

u subscript 2 equals b

u subscript k plus 2 end subscript equals u subscript k plus 1 end subscript over u subscript k

where a and b are real numbers.

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

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7b7 marks

Given that 

sum from r equals 1 to 44 of u subscript r space equals negative 50

sum from r equals 1 to 84 of u subscript r space equals negative 92

find the possible values of a and b.

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84 marks

Show that, for all n element of straight natural numbers,

         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 identical to sum from r equals 1 to 2 n of space r

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