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Question

# For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. Deposit amount: \$100; total deposits: 120; interest rate: 10%, compounded semi-annually

Solution

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The given annuity problem describes a geometric sequence with $a_1= 100$, $n= 120$ and $r= 1+\dfrac{.1}{2}=1.05$. Using $S_n=\dfrac{a_1(1-r^n)}{1-r}$ or the formula for a finite geometric series, then

\begin{align*} S_n&=\dfrac{a_1(1-r^n)}{1-r} \\\\ S_{120}&=\dfrac{100(1-(1.05)^{120})}{1-1.05} \\\\&\approx 695823.97 .\end{align*}

Hence, the total amount after $120$ deposits is $\695,823.97$.

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