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Let `f(x) = x+f(x-1)` where `xepsilonR`. If `F(0)=1` find `f(100)` |
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Answer» Given `f(x)=x+f(x-1) " and " f(0)=1` Put `x=1`. Then, `f(1)=1+f(0)=2` Put `x=2`. Then, `f(2)=2+f(1)=4` Put `x=3`. Then, `f(3)=3+f(2)=7` Thus, `f(0),f(1),f(2), …" form a series " 1,2,4,7, …. ` Let `S=1+2+4+7+ … +f(n-1)` `S=1+2+4+ ... +f(n-2)+f(n-1)` Subtracting , we get `0=(1+1+2+3+...+n" terms")-f(n-1)` ` :. f(n-1)=(n(n+1))/(2)` ` :. f(100)=5051` |
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