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Math Help - Prove 1-(1/n)<an<1

  1. #1
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    Prove 1-(1/n)<an<1

    Suppose the numbers a0,a1,a2,...,an satisfy the following conditions:
    a0=1/2, a(k+1)=ak+(1/n)(ak)^2 ; k=1,2,...,n-1
    Prove that 1-(1/n)<an<1.
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  2. #2
    Senior Member Sambit's Avatar
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    Calculate the value of a_1, check that 1-\frac{1}{n}<a_1<1 and use method of induction to prove the required.
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  3. #3
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    by induction,i have
    Let n=1, 0<an<1 (true)

    we assume n=k is true.
    1-(1/k)<ak<1

    Let n=k+1
    1-(1/k)+(1/n)(ak)^2<ak+(1/n)(ak)^2<1+(1/n)(ak)^2
    1-(1/k)+(1/k+1)(ak)^2<a(k+1)<1+(1/k+1)(ak)^2

    i can't solve for n=k+1
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