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Math Help - I want complete this answers >> help me

  1. #1
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    I want complete this answers >> help me

    Hi

    In like this question in
    I stop in same point and and I can't complete next part of question please help


    here I attached three question and I want complete the answers with explain

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    Attached Thumbnails Attached Thumbnails I want complete this answers >> help me-111.jpg   I want complete this answers >> help me-2222.jpg   I want complete this answers >> help me-333.jpg  
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  2. #2
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    Quote Originally Posted by r-soy View Post
    Hi

    In like this question in
    I stop in same point and and I can't complete next part of question please help


    here I attached three question and I want complete the answers with explain

    ------------
    Dear r-soy,

    In each of the problems you could simplify your last expressions further,

    1) \frac{k(k+1)(2k+1)}{6}+(k+1)^2

    (k+1)\left[\frac{k(2k+1)}{6}+(k+1)\right]

    (k+1)\frac{(2k^2+7k+6)}{6}

    \frac{(k+1)(k+2)(2k+3)}{6}

    \frac{(k+1)[(k+1)+1][2(k+1)+1]}{6}

    Therefore the statement is true for P(k+1).


    2) \frac{k(k+1)(k+2)}{3}+(k+1)(k+2)

    (k+1)(k+2)\left[\frac{k}{3}+1\right]

    (k+1)(k+2)\left(\frac{k+3}{3}\right)

    \frac{(k+1)[(k+1)+1][(k+1)+2]}{3}

    Therefore the statement is true for P(k+1).

    3) I think there is a bit of confusion in this problem. So I have done it from the beginning.

    P(n) : (xy)^n=x^{n}y^{n}

    P(1) : Left hand side = (xy)^1=xy

    Right hand side = x^{1}y^{1}=xy

    Therefore the statement is true for P(1).

    Suppose that the statement is true for P(k). Then,

    P(k) : (xy)^k=x^{k}y^{k}

    Therefore, (xy)^{k}(xy)=x^{k}y^{k}(xy)

    (xy)^{k+1}=x^{k+1}y^{k+1}

    Therefore the statement is true for P(1).

    Hope this will help you.
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  3. #3
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