Thread: 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.

Calculate the value of , check that 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