Problem $3$ Let $f(k)$ be a function where $k$ is an integer. We have seen that

${\displaystyle \underset{k=1}{\overset{N}{}}}(f(k)-f(k-1))=f(N)-f(0)$

Show that for any two function $f(k)$ and $g(k)$ we have

${\displaystyle \underset{k=1}{\overset{N}{}}}f(k)(g(k)-g(k-1))=[f(N)g(N)-f(0)g(0)]-{\displaystyle \underset{k=1}{\overset{N}{}}}(f(k)-f(k-1))g(k-1).$

$($Hint: Put both sums on the same side and simplify. For future reference: this can be seen as the "sum version" of integration by parts$)$

$2.$ Find a formula for ${\displaystyle \underset{k=1}{\overset{N}{}}}k{5}^k$ by applying the above formula for $f(k)=k$ and $g(k)=\frac{5^{k+1}}{4}.$