Question

At the beginning of year 1, Sam invests $700 at an annual compound interest

rate of 5%. He makes no deposits to or withdrawals from the account.
Which explicit formula can be used to find the account's
beginning of year 4? What is the balance?

Answer 1

at the beginning of year 4, only 3 years have elapsed, the 4th year hasn't started yet, since it's at the beginning, so at the beginning of year 4 we can say only 4-1 years have elapsed.

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$700\\ r=rate\to 5\%\to (5)/(100)\dotfill &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=\textit{elapsed years}\dotfill &3 \end{cases}`

`A=700\left( 1 + (0.05)/(1) \right)^(1\cdot 3)\implies A = 700(1+0.05)^3\implies A(4)=700(1+0.05)^(4-1) \\\\[-0.35em] ~\dotfill\\\\ ~\hfill A(n)=700(1+0.05)^(n-1)~\hfill`