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Question

Let f(x) be a differentiable function such that f(0)=1, and the sequence {an} is defined as a1=2 and an=limxx2(f(an1x)f(0))2 for n2. If the value of 10i=1ai=2k, then the value of k is

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Solution

an=limxx2(f(an1x)f(0))2
Putting x=an1h
If x, then h0
an=limh0(an1h)2(f(h)f(0))2=(an1)2limh0(f(h)f(0)h)2an=[an1f(0)]2an=(an1)2 (f(0)=1)

Now, an=(an1)2
Hence, a2=a21=22;
a3=a22=24;
a4=a23=28 and so on.
10i=1ai=2(1+2+22+23++29)=2 210121=21023
k=1023

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