You deposit \( \$ 500 \) each month into an account earning \( 5 \% \) interest compounded monthly. a) How much will you have in the account in 30 years? b) How much total money will you put into the account? c) How much total interest will you earn? \( \$ \) Question Help: Video 1 Video 2 Post to forum

To solve this problem, we will use the formula for the future value of a series of cash flows (an annuity) compounded at a certain interest rate. The formula is: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Where: - \( FV \) is the future value of the annuity. - \( P \) is the amount deposited each period (monthly). - \( r \) is the interest rate per period. - \( n \) is the total number of periods. ### Given: - Monthly deposit \( P = 500 \) - Annual interest rate \( = 5\% \) or \( 0.05 \) - Monthly interest rate \( r = \frac{0.05}{12} \) - Total number of years \( = 30 \) - Total number of months \( n = 30 \times 12 \) Now, let's calculate each part step by step. ### Step 1: Calculate \( r \) and \( n \) 1. Monthly interest rate: \[ r = \frac{0.05}{12} \] 2. Total number of months: \[ n = 30 \times 12 \] ### Step 2: Calculate the future value \( FV \) Using the formula for future value, we can substitute the values we calculated. ### Step 3: Calculate the total money deposited The total money deposited over 30 years is simply: \[ \text{Total Deposits} = P \times n \] ### Step 4: Calculate the total interest earned The total interest earned can be calculated as: \[ \text{Total Interest} = FV - \text{Total Deposits} \] Now, let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(30\times 12\) - step1: Multiply the numbers: \(360\) Calculate or simplify the expression \( 0.05/12 \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.05}{12}\) - step1: Convert the expressions: \(\frac{\frac{1}{20}}{12}\) - step2: Multiply by the reciprocal: \(\frac{1}{20}\times \frac{1}{12}\) - step3: Multiply the fractions: \(\frac{1}{20\times 12}\) - step4: Multiply: \(\frac{1}{240}\) Calculate or simplify the expression \( 500*360 \). Calculate the value by following steps: - step0: Calculate: \(500\times 360\) - step1: Multiply the numbers: \(180000\) Calculate or simplify the expression \( 500*((1+(0.004166666666666667))^360-1)/(0.004166666666666667) \). Calculate the value by following steps: - step0: Calculate: \(\frac{500\left(\left(1+\frac{4166666666666667}{1000000000000000000}\right)^{360}-1\right)}{\frac{4166666666666667}{1000000000000000000}}\) - step1: Add the numbers: \(\frac{500\left(\left(\frac{1004166666666666667}{1000000000000000000}\right)^{360}-1\right)}{\frac{4166666666666667}{1000000000000000000}}\) - step2: Subtract the numbers: \(\frac{500\times \frac{1004166666666666667^{360}-1000000000000000000^{360}}{1000000000000000000^{360}}}{\frac{4166666666666667}{1000000000000000000}}\) - step3: Multiply the numbers: \(\frac{\frac{1004166666666666667^{360}-1000000000000000000^{360}}{500^{2159}\times 64^{360}}}{\frac{4166666666666667}{1000000000000000000}}\) - step4: Multiply by the reciprocal: \(\frac{1004166666666666667^{360}-1000000000000000000^{360}}{500^{2159}\times 64^{360}}\times \frac{1000000000000000000}{4166666666666667}\) - step5: Rewrite the expression: \(\frac{1004166666666666667^{360}-1000000000000000000^{360}}{500^{2159}\times 64^{360}}\times \frac{15625000000000000\times 64}{4166666666666667}\) - step6: Rewrite the expression: \(\frac{1004166666666666667^{360}-1000000000000000000^{360}}{500^{2159}\times 64^{360}}\times \frac{500^{6}\times 64}{4166666666666667}\) - step7: Reduce the numbers: \(\frac{1004166666666666667^{360}-1000000000000000000^{360}}{500^{2153}\times 64^{360}}\times \frac{64}{4166666666666667}\) - step8: Rewrite the expression: \(\frac{1004166666666666667^{360}-1000000000000000000^{360}}{4^{2153}\times 125^{2153}\times 64^{360}}\times \frac{64}{4166666666666667}\) - step9: Rewrite the expression: \(\frac{1004166666666666667^{360}-1000000000000000000^{360}}{4^{2153}\times 125^{2153}\times 64^{360}}\times \frac{4^{3}}{4166666666666667}\) - step10: Reduce the numbers: \(\frac{1004166666666666667^{360}-1000000000000000000^{360}}{4^{2150}\times 125^{2153}\times 64^{360}}\times \frac{1}{4166666666666667}\) - step11: Multiply the fractions: \(\frac{1004166666666666667^{360}-1000000000000000000^{360}}{4^{3230}\times 125^{2153}\times 4166666666666667}\) - step12: Multiply: \(\frac{1004166666666666667^{360}-1000000000000000000^{360}}{4166666666666667\times 4^{3230}\times 125^{2153}}\) Calculate or simplify the expression \( 416129.317681-180000 \). Calculate the value by following steps: - step0: Calculate: \(416129.317681-180000\) - step1: Subtract the numbers: \(236129.317681\) Now that we have performed the calculations, let's summarize the results for each part of the problem. ### a) Future Value in the Account After 30 Years Using the future value formula, we found: \[ FV \approx 416129.32 \] So, you will have approximately **\$416,129.32** in the account after 30 years. ### b) Total Money Deposited The total amount of money you will deposit into the account over 30 years is: \[ \text{Total Deposits} = 500 \times 360 = 180000 \] Thus, the total money deposited is **\$180,000**. ### c) Total Interest Earned The total interest earned can be calculated as: \[ \text{Total Interest} = FV - \text{Total Deposits} \approx 416129.32 - 180000 \approx 236129.32 \] Therefore, the total interest earned is approximately **\$236,129.32**. ### Summary of Results: - **Future Value in the Account:** \$416,129.32 - **Total Money Deposited:** \$180,000 - **Total Interest Earned:** \$236,129.32