Chapter 10, Problem 75E

Linear relationship between $V$ and $P$ in the retirement savings formula. [In all the retirement savings accounts, assume that the frequency of contributions (annual or monthly), the term, and the APR all remain the same.]

a. Explain why, in the retirement savings formula (be it the annual or the monthly version), the value $V$ of the account is proportional to the contribution $P$. In other words, explain why if a contribution of $P$ generates an account with value $V$, then a contribution of $cP$ (where $c$ is any positive constant) generates an account with value $V$. (Hint: What happens in the retirement savings formula when you replace $P$ by $cP$?)

b. Explain why if a contribution of $P$ generates a retirement savings account with value $V$. and a contribution of $Q$ generates a retirement savings account with value $W$, then a contribution of $\left(P+Q\right)$ generates a retirement savings account with value $\left(V+W\right)$.