Question

Find the future value of an ordinary annuity if payments are made in the amount r and interest is compounded as given.then determine how much of this value is from contribution and how much is from interest

r=11,000;4.1% interest compounded quarterly for 13 years

Answer 1

`~~~~~~~~~~~~\stackrel{\textit{payments at the beginning of the period}}{\textit{Future Value of an annuity due}} \\\\ A=pmt\left[ \cfrac{\left( 1+(r)/(n) \right)^(nt)-1}{(r)/(n)} \right]\left(1+(r)/(n)\right)`

`\qquad \begin{cases} A=\textit{accumulated amount} \\ pmt=\textit{periodic payments}\dotfill & 11000\\ r=rate\to 4.1\%\to (4.1)/(100)\dotfill &0.041\\ n= \begin{array}{llll} \textit{times it compounds per year} \end{array}\dotfill &4\\ t=years\dotfill &13 \end{cases}`

`A=11000\left[ \cfrac{\left( 1+(0.041)/(4) \right)^(4 \cdot 13)-1}{(0.041)/(4)} \right]\left(1+(0.041)/(4)\right) \\\\\\ A=11000\left[ \cfrac{\left( 1.01025 \right)^(4 \cdot 13)-1}{0.01025} \right]\left(1.01025\right) \implies A \approx 758290.48 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{contribution from depositor}}{(11000)(4)(13)\implies \text{\LARGE 572000}}\hspace{5em}\underset{ earned~interest }{\stackrel{ 758290.48~~ - ~~572000 }{\text{\LARGE 186290.48}}}`