General Binomial Expansion (CIE A Level Maths: Pure 3)

Find, in ascending powers of , the binomial expansion of

up to and including the term in .

Find the first three terms, in ascending powers of , in the binomial expansion of

State the values of  for which your expansion in part (a) is valid.

Using a suitable value of , use your expansion from part (a) to estimate , giving your answer to 3 significant figures.

Find, in ascending powers of , the binomial expansion of

up to and including the term in .

Use the binomial expansion to show that the first three terms in the expansion of  are  

Hence, or otherwise, find the expansion of   up to and including the term in .

The function  is given by

where  is an integer.

Find the coefficient of the term in  in the binomial expansion of  , in terms of .

Find the coefficient of the term in  in the binomial expansion of  , in terms of .

In the binomial expansion of  the coefficient of the term in  is equal to the coefficient of the term in .
Find the value of .

The functions  and  are given as follows

Expand , in ascending powers of  up to and including the term in .

Find the values for  for which the expansion is valid.

In the expansion of  , where  is a negative integer, the coefficient of the term in  is .

Find the value of .

Express   in partial fractions.

Use the binomial expansion to find the first three terms, in ascending powers of , in each of   and 

Hence show that 

Write down the values of  for which your expansion in part (c) converges.

Given that  is small such that and higher powers of  can be ignored show that

For which values of  is the approximation in part (a) valid?

 It is given that

                  and       

In their binomial expansions, the coefficient of the  term for is equal to the coefficient of the term for 

Find the value of .

Express   in partial fractions.

Using binomial expansions, up to and including terms in  show that

Explain why the approximation in part (b) is only valid for

Find the first three terms in the binomial expansion of  

Show that the expansion is valid for 

The expansion is to be used in a computer program to estimate the value of  .
Find the value of  to be used and check it meets the validity requirement from part (b).

Hence find the value the computer program will use to estimate  .

Find, in ascending powers of , the binomial expansion of

up to and including the term in .

Use the first three terms, in ascending powers of  in the binomial expansion of

to estimate the value of , giving your answer to three significant figures.

Explain why your approximation in part (a) is valid.

Find, in ascending powers of , the binomial expansion of

up to and including the term in .

Use the binomial expansion to expand   up to and including the term in 

Hence, or otherwise, expand   up to and including the term in .

In the expansion of    the coefficient of the term in  is double the coefficient of the term in .  Find the value of  .

The functions and   are given as follows

Expand , in ascending powers of  up to and including the term in .

Expand , in ascending powers of  up to and including the term in .

Find the expansion of   in ascending powers of , up to and including the term in .

Find the values of  for which your expansion in part (c) is valid.

In the expansion of   , where  is a real number, the coefficient of the term in is .

Find the possible values of .

Express   in partial fractions.

Use the binomial expansion to find the first three terms, in ascending powers of , in each of , , and .

Hence express   as the first three terms of a binomial expansion in ascending powers of  .

Write down the values of  for which your expansion in part (c) converges.

Given that  is small such that and higher powers of  can be ignored show that

Find the percentage error between your calculator answer and the approximation in part (a) when , giving your answer to one decimal place.

For which values of  is the approximation in part (a) valid?

In the binomial expansion of     where , the coefficient of the  term is equal to the coefficient of the  term.

Show that .

Given further that  find the values of and .

Express   in the form  , where  and  are integers to be found.

Hence, or otherwise, find the binomial expansion of , in ascending powers of , up to and including the term in .

The expansion in part (b) is to be used to approximate the value of a fraction.

If , which fraction is being approximated?

Find the first four terms in the binomial expansion of  

Find the values of  for which the expansion is valid.

The expansion is to be used in a computer program to estimate the value of .
Check that the expansion is valid for this purpose and use the first four terms of the expansion to estimate the value of .

Find the percentage error the computer program will introduce by using the expansion as an approximation to  

Find, in ascending powers of , the binomial expansion of

up to and including the term in .

Use the first three terms, in ascending powers of , in the binomial expansion of

to estimate the value of , giving your answer to two decimal places.

Explain why you would not be able to use your expansion to approximate .

Find, in ascending powers of , the binomial expansion of

up to and including the term in .

Expand  up to and including the term in .

In the expansion of   ,  the coefficient of the term in  is one-seventh of the coefficient of the term in .  Find the value of .

The functions  and  are given as follows

Find the binomial expansion of   , in ascending powers of , up to and including the term in .  Also find the values of for which your expansion is valid.

In the expansion of  , where  is a real number, the coefficient of the term in  is .

Given that  find the value of .

Express    in partial fractions.

Express   as the first three terms of a binomial expansion in ascending powers of .

Write down the values of  for which your expansion in part (b) converges.

Given that  is small such that and higher powers of  can be ignored show that

         

Find the percentage error between your calculator answer and the approximation in part (a) when , giving your answer to one decimal place.

For which values of  is the approximation in part (a) valid?

It is given that

                 and              

The binomial expansions of    and  have the following properties:

The coefficient of the  term in the expansion of  is 72 times larger than the coefficient of the  term in the expansion of .

The coefficient of the  term in the expansion of  is 24 times larger than the coefficient of the  term in the expansion of .

Find the values of  and .

Find the binomial expansion of , in ascending powers of , up to and including the term in .

Explain why the expansion found in part (a) cannot be used when .

The binomial expansion of

is to be used in a computer program to estimate the reciprocal of .

The computer program needs to be accurate to at least 5 significant figures when compared to the value produced by a scientific calculator.

Find the least number of terms from the expansion that are required for the computer program.  Justify that the expansion used is valid.