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Let f(x)=x+f(x-1) for AAx in R. If f(0)...

Let `f(x)=x+f(x-1)` for `AAx in R`. If `f(0)=1,f i n d \ f(100)`.

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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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