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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


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The given annuity problem describes a geometric sequence with a1=100a_1= 100, n=120n= 120 and r=1+.12=1.05r= 1+\dfrac{.1}{2}=1.05. Using Sn=a1(1rn)1rS_n=\dfrac{a_1(1-r^n)}{1-r} or the formula for a finite geometric series, then

Sn=a1(1rn)1rS120=100(1(1.05)120)11.05695823.97.\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 120120 deposits is $695,823.97\$695,823.97.

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