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Find the product

Posted on: October 13th, 2014 by

What is (1-1/4)(1-1/9)(1-1/16)....(1-1/n^2)?

- via Mind your decision

2 Responses to Find the product

  1. superman had this to say about that:

    1-1/n^2 = (n^2-1)/n^2 = (n-1)(n+1)/n^2

    (1-1/(n-1)^2)(1-1/n^2) = (n-2)(n)/(n-1)^2 * (n-1)(n+1)/n^2 = (n-2)(n+1)/((n-1)n)

    Required answer = 1*(n+1)/(2*n)

  2. Dinesh had this to say about that:

    The general term is \{ (1-\frac{1}{n^2}) \} which is the same as \{ \frac{(n-1)(n+1)}{n^2} \}. Separating into two products, we get

     \frac{1}{2} \dots \frac{N-1}{N} \times \frac{3}{2} \dots \frac{N+1}{N}

    which after telescopic cancellation gives \{ \frac{N+1}{2N} \}

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