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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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.
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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