Sequences & Series (Edexcel International A Level Maths: Pure 2)

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 , 

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 .

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     

Calculate the value of .

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  , 

(i)   show that      

(ii)   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.

Prove that, for all ,