Let's solve the problem step-by-step. ### **Given:** - **Periodic Deposit (\(PMT\))**: $70 (at the end of each month) - **Rate (\(r\))**: 5% compounded monthly - **Time (\(t\))**: 15 years ### **Part a: Find the value of the annuity (\(F\))** The formula for the future value of an ordinary annuity (deposits at the end of each period) is: \[ F = PMT \times \frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}} \] Where: - \( PMT = \$70 \) - \( r = 0.05 \) (annual rate) - \( n = 12 \) (compounded monthly) - \( t = 15 \) years #### **Step 1: Plug in the values** \[ F = 70 \times \frac{(1 + \frac{0.05}{12})^{12 \times 15} - 1}{\frac{0.05}{12}} \] #### **Step 2: Calculate \(\frac{0.05}{12}\)** \[ \frac{0.05}{12} = 0.0041667 \] #### **Step 3: Calculate \(12 \times 15 = 180\)** #### **Step 4: Calculate \(1 + 0.0041667 = 1.0041667\)** #### **Step 5: Raise to the power of 180** \[ (1.0041667)^{180} \] Using a calculator: \[ (1.0041667)^{180} \approx 2.11357 \] #### **Step 6: Subtract 1** \[ 2.11357 - 1 = 1.11357 \] #### **Step 7: Divide by 0.0041667** \[ \frac{1.11357}{0.0041667} \approx 267.34 \] #### **Step 8: Multiply by 70** \[ 70 \times 267.34 = 18,713.8 \] #### **Step 9: Round to the nearest dollar** \[ \boxed{\$18,714} \] --- ### **Part b: Find the Interest** Interest earned is the difference between the future value and the total deposits. \[ \text{Total deposited} = PMT \times n \times t = 70 \times 12 \times 15 = 70 \times 180 = \$12,600 \] \[ \text{Interest earned} = F - \text{Total deposited} = 18,714 - 12,600 = \boxed{\$6,114} \] --- ## **Final Answers** **a. The value of the annuity is** **\$18,714.** **b. The interest is** **\$6,114.**