Question 6 Suppose an account pays \( 5 \% \) interest that is compounded annually. At the beginning of each year, \( \$ 1,000 \) is deposited into the account (starting with \( \$ 1,000 \) for the first year). How much money is in the account at the end of the tenth year if no withdrawals or additional deposits are made?

To determine the amount of money in the account at the end of the tenth year, we'll analyze the contributions and the interest earned over the period. Here's a step-by-step breakdown: ### Given: - **Annual Interest Rate (r):** 5% or 0.05 - **Annual Deposit (P):** \$1,000 - **Number of Years (n):** 10 - **Type of Annuity:** **Annuity Due** (since deposits are made at the **beginning** of each year) ### Understanding Annuity Due: An **annuity due** means that each payment is made at the start of each period, allowing each deposit to earn interest for an additional period compared to an ordinary annuity. ### Future Value Formula for Annuity Due: The future value \( FV \) of an annuity due can be calculated using the formula: \[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \times (1 + r) \] ### Plugging in the Numbers: 1. **Calculate \( (1 + r)^n \):** \[ (1 + 0.05)^{10} = 1.6288946268 \] 2. **Calculate \( (1.6288946268 - 1) / 0.05 \):** \[ \frac{0.6288946268}{0.05} = 12.577892536 \] 3. **Multiply by \( (1 + r) \):** \[ 12.577892536 \times 1.05 = 13.206787163 \] 4. **Multiply by the annual deposit \( P \):** \[ 13.206787163 \times \$1,000 = \$13,206.79 \] ### Final Amount: After ten years, the account will contain approximately **\$13,207**. ### Summary: - **Initial Deposit:** \$1,000 at the beginning of each year - **Interest Rate:** 5% compounded annually - **Duration:** 10 years - **Final Amount:** **\$13,207**