Find, in ascending powers of , the binomial expansion of
        Â
up to and including the term in .
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Find, in ascending powers of , the binomial expansion of
        Â
up to and including the term in .
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Find the first three terms, in ascending powers of , in the binomial expansion of
      Â
State the values of for which your expansion in part (a) is valid.
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Show that
        Â
Hence find, in ascending powers of , the first three terms in the binomial expansion of
    Â
Using , use your expansion from part (b) to find an approximation to  .
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Find, in ascending powers of , the binomial expansion of
      Â
up to and including the term in .
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Find the first three terms, in ascending powers of , in the binomial expansion of
    Â
State the values of for which your expansion in part (a) is valid.
Â
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Find the coefficient of the term in  in the binomial expansion of
      Â
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The function  is given by
       Â
where  is an integer.
Find the coefficient of the term in  in the binomial expansion of , in terms of .
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Given that  is small such that and higher powers of can be ignored show that
          Â
Using a suitable value of in the result from part (a), find an approximation for the value of .
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It is given that
              Â
where  is a non-zero constant.
In their binomial expansions, the coefficient of the  term for  is equal to the coefficient of the term forÂ
Find the value of .
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Show, as partial fractions, that
        Â
Find the first three terms, in ascending powers of , of the binomial expansion of
Hence show that the first three terms, in ascending powers of , in the binomial expansion of
           Â
are
              Â
Write down the values of for which this expansion converges.
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Find, in ascending powers of , the binomial expansion of
        Â
up to and including the term in .
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Find the first three terms, in ascending powers of , in the binomial expansion of
               Â
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.
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Find, in ascending powers of , the binomial expansion of
           Â
up to and including the term in .
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Use the binomial expansion to show that the first three terms in the expansion of are Â
Hence, or otherwise, find the expansion of  up to and including the term in .
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The function  is given by
           Â
where  is an integer.
Â
In the binomial expansion of the coefficient of the term in is equal to the coefficient of the term in .
Find the value of .
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The functions  and  are given as follows
        Â
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In the expansion of , where  is a negative integer, the coefficient of the term in  is .
Find the value of .
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Express  in partial fractions.
Use the binomial expansion to find the first three terms, in ascending powers of , in each of  andÂ
Hence show thatÂ
Write down the values of for which your expansion in part (c) converges.
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Given that  is small such that and higher powers of can be ignored show that
      Â
For which values of is the approximation in part (a) valid?
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 It is given that
              and   Â
                 Â
In their binomial expansions, the coefficient of the  term for is equal to the coefficient of the term forÂ
Find the value of .
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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
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Find, in ascending powers of , the binomial expansion of
         Â
up to and including the term in .
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Use the first three terms, in ascending powers of in the binomial expansion of
            Â
       Â
to estimate the value of , giving your answer to three significant figures.
Explain why your approximation in part (a) is valid.
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Find, in ascending powers of , the binomial expansion of
        Â
up to and including the term in .
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Use the binomial expansion to expand  up to and including the term inÂ
Hence, or otherwise, expand  up to and including the term in .
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In the expansion of  the coefficient of the term in  is double the coefficient of the term in . Find the value of .
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The functions and   are given as follows
             Â
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.
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In the expansion of  , where  is a real number, the coefficient of the term in is .
Find the possible values of .
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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.
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Given that  is small such that and higher powers of can be ignored show that
       Â
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?
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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 .
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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.
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Find, in ascending powers of , the binomial expansion of
        Â
up to and including the term in .
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Use the first three terms, in ascending powers of , in the binomial expansion of
        Â
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 .
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Find, in ascending powers of , the binomial expansion of
        Â
up to and including the term in .
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Expand up to and including the term in .
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In the expansion of  , the coefficient of the term in  is one-seventh of the coefficient of the term in . Find the value of .
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The functions  and  are given as follows
                Â
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.
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In the expansion of , where  is a real number, the coefficient of the term in  is .
Given that find the value of .
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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.
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Given that  is small such that and higher powers of can be ignored show that
        Â
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?
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It is given that
             and       Â
The binomial expansions of  and have the following properties:
Â
Find the values of  and .
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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 .
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