The task is to use index shifting on the following sum and product to show the equality but it doesn't get me there.
(1) $\displaystyle\sum^n_{k=m} a_k - a_{k-1} = a_n - a_{m-1}$
and also for
(2) $\displaystyle\prod^{n}_{k=m} \dfrac{a_k}{a_{k-1}} = \dfrac{a_n}{a_{m-1}}$
(1) $$\sum_{k=m}^n (a_k-a_{k-1})=\sum_{k=m}^n a_k-\sum_{k=m}^n a_{k-1}=\sum_{k=m}^n a_k-\sum_{k=m-1}^{n-1} a_k=\sum_{k=m}^n a_k-\left[\sum_{k=m}^n a_k-a_{n}+a_{m-1}\right]$$ $$=\sum_{k=m}^n a_k-\sum_{k=m}^n a_k+a_{n}-a_{m-1}=a_n-a_{m-1}$$ Same method for (2)