Question

Please helpppppp

Anthony deposits $2,000 twice a year into his IRA account. If this account earns 6. 5% compounded

semi-annually, how much will Anthony have when he retires in 35 years? How much interest will

Anthony have earned over the 35 years?

Answer 1

now, we're making an assumption he's making those deposits at the start of the period, not in the end, so

`~~~~~~~~~~~~\stackrel{\textit{deposits 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 deposits}\dotfill & 2000\\ r=rate\to 6.5\%\to (6.5)/(100)\dotfill &0.065\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{semi-annually, thus twice} \end{array}\dotfill &2\\ t=years\dotfill &35 \end{cases}`

`A=2000\left[ \cfrac{\left( 1+(0.065)/(2) \right)^(2 \cdot 35)-1}{(0.065)/(2)} \right]\left(1+(0.065)/(2)\right) \\\\\\ A=2000\left[ \cfrac{\left( 1.0325 \right)^(70)-1}{0.0325} \right]\left(1.0325\right) \implies \boxed{A \approx 532591.46} \\\\[-0.35em] ~\dotfill\\\\ 532591.46~~ - ~~\stackrel{ deposits~made }{(2)(2000)(35)} ~~ \approx ~~ \stackrel{ earned~interest }{\boxed{252591.46}}`