if $f(n)=G(n)-G(n-1)$ then, prove if $f(1)+\cdots+f(n)=G(n)-G(0)$ is true via induction

Question

I have a problem where I have to prove if $f(1)+\cdots+f(n)=G(n)-G(0)$ is true via induction, knowing $f(n)=G(n)-G(n-1)$.

I´ve tried to begin with this problem by myself but I don´t really know how to solve it.

Thanks a lot!

Answer 1

You probably mean

$$f(1) + \ldots f(n)=G(n)-G(0)\tag{$n$}$$

when given (for $n \ge 1$) that

$$f(n)=G(n)-G(n-1)\tag{*}$$

So to see the statement for $n=1$, just apply (*) for $n=1$.

Assume ($n$) holds, and

$$f(1)+ \ldots f(n)+f(n+1) = (f(1)+ \ldots f(n)) + f(n+1) = G(n)-G(0) +( G(n+1)-G(n))= \\G(n+1)-G(0)$$

showing ($n+1$), where we apply (*) for $n+1$ as well.