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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- Mar 15th 2011, 06:24 PMchris86Prove 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. - Mar 16th 2011, 03:48 AMSambit
Calculate the value of $\displaystyle a_1$, check that $\displaystyle 1-\frac{1}{n}<a_1<1$ and use method of induction to prove the required.

- Mar 16th 2011, 05:15 AMchris86
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