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

The first terms of a series are given by .

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

Given that    ,

Show that 

Hence find the value of .

The first  terms of a series are given by  .

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

Given that  = 20470,

Show that 

For this value of , calculate .

A geometric series is given by 

Write down the common ratio, , of the series.

Given that the series is convergent, and that , calculate the value of .

An arithmetic series is given by 

Given that  and , find the values of  and .

A sequence is defined for by the recurrence relation  

Calculate

A sequence is defined for by the recurrence relation 

Calculate, giving your answers as exact values 

A sequence is defined for by the recurrence relation 

where  is a constant.

Write down expressions for  and   in terms of .

Given that the sequence is periodic with order 2, and given as well that ,

Find the value of .

For the value of  found in part (b)

Calculate 

The terms of a sequence are defined by  for all .

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

It can be shown that, for all ,

Using that formula,

Calculate 

Find the value of  ,i.e. the sum of the squares of all the integers between 51 and 100 inclusive.

Given that , 

Show that 

Hence, find the value of .

Given that  and , find the values of  and 

Given that ,

Show that 

For this value of , calculate .

A convergent geometric series is given by 

Write down the range of possible values of .

Given that  =24

Calculate the value of .

A sequence is defined for by the recurrence relation  

Calculate

A sequence is defined for by the recurrence relation  

Calculate, giving your answers as exact values

A sequence is defined for by the recurrence relation 

where  is a constant.

Given that the sequence is periodic with order 2, and given as well that ,

Find the value of  .

For the value of  found in part (a),

Calculate  

The terms of a sequence are defined, for all   by  .

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

It can be shown that, for all ,

            =     and             

Using those formulas,

Show that .

Given that  , find the value of .

Given that ,  

show that     

hence find the value of .

For this value of , calculate .

Given that ,  and  , find the values of  and .

A convergent geometric series is given by , 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 .

Given that 

Calculate the value of  .

A sequence is defined for by  .

Calculate , giving your answer as an exact value.

A sequence is defined for all  by

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

A different sequence is defined for all by 

where  is a real constant.

Given that the sequence is not periodic,

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

A sequence is defined for by the recurrence relation  

where  and  are real numbers.

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

Given that  and   

determine the possible values of  and .

Prove that, for all ,