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The accumulated value of deposits $\$1140$ at the end of every $6$ months for four and one-half years when interest is $12\%$ compounded annually

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Exponential growth and decay: word problems

Full solution

Q. The accumulated value of deposits

\$1140

at the end of every

6

months for four and one-half years when interest is

12\%

compounded annually

Identify compounding periods and deposits: Identify the number of compounding periods per year and the total number of deposits. $\newline$ Compounding periods per year: $1$ (since it's compounded annually) $\newline$ Total number of deposits: $4.5$ years $\times 2$ deposits/year $= 9$ deposits
Calculate interest rate per period: Calculate the interest rate per compounding period. $Annual interest rate: 12\%$ $Interest rate per compounding period: 12\%$ (since it's compounded annually)
Use annuity formula: Use the future value of an annuity formula: $FV = P \times \left[(1 + r)^n - 1\right] / r$ Where $P =$ periodic deposit, $r =$ interest rate per period, $n =$ total number of deposits. $P = \$1140$ , $r = 0.12$ , $n = 9$
Substitute values and calculate: Substitute the values into the formula and calculate the future value. $\newline$ $FV = 1140 \times \left[(1 + 0.12)^9 - 1\right] / 0.12$
Calculate future value: Calculate the future value. $\newline$ $FV = 1140 \times [(1.12)^9 - 1] / 0.12$ $\newline$ $FV = 1140 \times [2.7731 - 1] / 0.12$ $\newline$ $FV = 1140 \times 1.7731 / 0.12$ $\newline$ $FV = 2021.334 / 0.12$ $\newline$ $FV = 16844.45$

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