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Math Help - Binomial expansion

  1. #1
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    Binomial expansion

    In each of the following expansions , find the term as stated .

    (1) (1+x)^{10} , 5th term

    (2) (2a+b)^{12} , 10th term

    (3) (x-\frac{1}{x})^6 , constant term
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  2. #2
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    Quote Originally Posted by thereddevils View Post
    In each of the following expansions , find the term as stated .

    (1) (1+x)^{10} , 5th term

    (2) (2a+b)^{12} , 10th term

    (3) (x-\frac{1}{x})^6 , constant term
    Use the formula:

    (a+b)^n=\sum_{k=0}^n {n \choose k}  a^{n-k} \cdot b^k

    Keep in mind that you start counting by zero! Thus the 5th term correspond with k = 4.

    Therefore the 5th term of (1+x)^{10} \rightarrow {10\choose 4}1^4\cdot x^6 = 210x^6<br />

    For the last example: A constant term only occurs if the exponents of x and \frac1x are equal, thus you have to calculate the 4th term with k = 3. Since k is an odd number you get as the constant summand -20.
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  3. #3
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    Quote Originally Posted by earboth View Post
    Therefore the 5th term of (1+x)^{10} \rightarrow {10\choose 4}1^4\cdot x^6 = 210x^6<br />

    I guess there is a slight typo , should be 210x^4
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  4. #4
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    Quote Originally Posted by mathaddict View Post
    I guess there is a slight typo , should be 210x^4
    With (1+x)^{10} the 5th term is:

     {10\choose 4}1^4\cdot x^6 = 210x^6 and the 7th term is  {10\choose 6}1^6\cdot x^4 = 210x^4<br />

    Mostly the binoms are given as (x+1)^{10} and then you would have been right.
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