How proof this with induction.

Question

I'm trying to address this exercise but do not know how to approach it:

if $f(n) = G(n)-G(n-1)$ for all $n \geq 1$, prove that $f(1) + f(2) + f(3) + \cdots + f(n) = G(n)-G(0)$ for all $n \geq 1$.

Any ideas? Thanks.

Answer 1

If $n = 1$, then by definition we have $\sum_{k=1}^{n}f(k) = f(1) = G(1) - G(0) = G(n) - G(0)$; if $n \geq 1$ such that $\sum_{k=1}^{n}f(k) = G(n) - G(0)$, then $$ \sum_{k=1}^{n+1}f(k) = G(n) - G(0) + f(n+1) = G(n) - G(0) + G(n+1) - G(n) = G(n+1) - G(0); $$ by mathematical induction we thus are done.