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Math Help - How do I solve this expansion?

  1. #1
    miket
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    How do I solve this expansion?

    I have two questions which I don't know how to solve... I really need any help you can give. They're both on expansions, which I missed due to being absent for a week. I would love it if you could give me the answer, and explain how you got it. Thanks



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  2. #2
    Grand Panjandrum
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    65. This is a question about the binomial expansion.

    You need to know that

    (a+b)^n = sum_{r=0 to n} [n!/((n-r)! r!] a^{n-r} b^r

    so in your case a=x^3, b=-y^2, n=20, and as you are asked for the 8-th
    term r=7.

    So the 8-th term is:

    [20!/(13! 7!)] (x^3)^13 (-y^2)^7

    which is answer D.

    RonL
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  3. #3
    Grand Panjandrum
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    68. This is again a binomial expansion problem. So again we are interested in:

    (a+b)^n = sum_{r=0 to n} [n!/((n-r)! r!] a^{n-r} b^r

    but in this case a=1, b=2x and n=6, and we are interested in the first four terms:

    6!/(6! 0!) 1^6 (2x)^0 + 6!/(5! 1!) 1^5 (2x)^1 + 6!/(4! 2!) 1^4 (2x)^2 + 6!/(3! 3!) 1^3 (2x)^3 = 1 + 6 (2x) + 15 (2x)^2 + 20 (2x)^3

    ......................= 1 + 12x + 60x^2 + 160x^3

    Which is answer J.

    RonL
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