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Question

If f(x+1)+f(x1)=2f(x) and f(0)=0, then f(n),nN, is

A
nf(1)
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B
{f(1)n}
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C
0
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D
None of these
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Solution

The correct option is A nf(1)
f(x1)+f(x+1)=2f(x) ...(it is given that f(0)=0)

Substituting x=1, we get

f(0)+f(2)=2f(1)

f(2)=2f(1) ...(f(0)=0)

Substituting x=2

f(1)+f(3)=2f(2)

f(1)+f(3)=4f(1) ...(f(2)=2f(1))

f(3)=3f(1)
:
:
Therefore f(n)=n(f(1))

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