QuestionJuly 21, 2025

10. Suppose you had 10000 in your account on 1st January 2006. The annual interest rate is 4% . You agreed to deposit a fixed amount K each year for eight years, the first deposit on 1st January 2009. What choice of the fixed amount K will imply that you have a balance of 70000 immediately after the last deposit?

10. Suppose you had 10000 in your account on 1st January 2006. The annual interest rate is 4% . You agreed to deposit a fixed amount K each year for eight years, the first deposit on 1st January 2009. What choice of the fixed amount K will imply that you have a balance of 70000 immediately after the last deposit?
10. Suppose you had 10000 in your account on 1st January 2006. The annual interest rate is 4% . You agreed to deposit a fixed amount K each year for eight years, the first deposit on 1st January 2009. What choice of the fixed amount K will imply that you have a balance of 70000 immediately after the last deposit?

Solution
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Answer

K \approx 4888.54 Explanation 1. Calculate Future Value of Initial Amount Use the formula for compound interest: FV = PV \times (1 + r)^n. Here, PV = 10000, r = 0.04, and n = 11 (from 2006 to 2017). Calculate FV = 10000 \times (1.04)^{11}. 2. Calculate Future Value of Annuity Deposits Use the future value of an annuity formula: FV_{\text{annuity}} = K \times \frac{(1 + r)^n - 1}{r}. Here, K is unknown, r = 0.04, and n = 9 (from 2009 to 2017). 3. Set Up Equation for Total Future Value The total future value should be 70000: FV_{\text{initial}} + FV_{\text{annuity}} = 70000. Substitute the values from Steps 1 and 2 into this equation. 4. Solve for K Rearrange the equation to solve for K: K = \frac{70000 - FV_{\text{initial}}}{\frac{(1.04)^9 - 1}{0.04}}. 5. Perform Calculations Compute FV_{\text{initial}} = 10000 \times (1.04)^{11} \approx 14802.44. Then compute K = \frac{70000 - 14802.44}{\frac{(1.04)^9 - 1}{0.04}}.

Explanation

1. Calculate Future Value of Initial Amount<br /> Use the formula for compound interest: $FV = PV \times (1 + r)^n$. Here, $PV = 10000$, $r = 0.04$, and $n = 11$ (from 2006 to 2017). Calculate $FV = 10000 \times (1.04)^{11}$.<br /><br />2. Calculate Future Value of Annuity Deposits<br /> Use the future value of an annuity formula: $FV_{\text{annuity}} = K \times \frac{(1 + r)^n - 1}{r}$. Here, $K$ is unknown, $r = 0.04$, and $n = 9$ (from 2009 to 2017).<br /><br />3. Set Up Equation for Total Future Value<br /> The total future value should be $70000$: $FV_{\text{initial}} + FV_{\text{annuity}} = 70000$. Substitute the values from Steps 1 and 2 into this equation.<br /><br />4. Solve for K<br /> Rearrange the equation to solve for $K$: $K = \frac{70000 - FV_{\text{initial}}}{\frac{(1.04)^9 - 1}{0.04}}$.<br /><br />5. Perform Calculations<br /> Compute $FV_{\text{initial}} = 10000 \times (1.04)^{11} \approx 14802.44$. Then compute $K = \frac{70000 - 14802.44}{\frac{(1.04)^9 - 1}{0.04}}$.
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