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This problem involves finding the present value of an annuity. The situation involves 25 total payments of $40,000 each, with one payment made immediately and 24 additional payments made at the end of each year for the next 24 years. The interest rate is 3% per year, compounded yearly.

To solve this, we use the formula for the present value of an annuity, specifically an **ordinary annuity**, but we'll adjust for the first payment being made immediately (which is not discounted).

### The present value $ PV $ of an ordinary annuity is given by:
\[
PV = P \times \frac{1 - (1 + r)^{-n}}{r}
\]
Where:
- $ P $ is the annual payment (\$40,000),
- $ r $ is the interest rate per period (3% = 0.03),
- $ n $ is the number of periods (24 years in this case).

Since the first payment is immediate, we'll add that first payment directly (without discounting), and then calculate the present value of the remaining 24 payments.

Let's now calculate this.The lottery commission should have approximately **$717,421.68** in the bank initially to guarantee the 25 annual payments of $40,000, assuming an interest rate of 3% compounded yearly.

Would you like further details on how the calculation was done or have any questions?

Here are five related questions you might explore:
1. What happens to the present value if the interest rate changes to 5%?
2. How would the present value differ if payments were made at the beginning of each year (annuity due)?
3. What would be the present value if the number of payments increased to 30 years?
4. How does the present value change if payments increase by a fixed percentage each year (growing annuity)?
5. How would inflation affect the purchasing power of these payments over 24 years?

**Tip**: Understanding the relationship between interest rates and present value can help you make informed financial decisions. A higher interest rate decreases the present value of future payments!

A state lottery commission pays the winner 25 installments of $40,000 per year. The commission makes the first payment of $40,000 immediately and the other 24 payments at the end of each of the next 24 years. Determine how much money the commission should have in the bank initially to guarantee the payments, assuming that the balance on deposit with the bank earns interest at the rate of 3% per year compounded yearly.

Calculate the present value of a lottery payout made in 25 installments, with an interest rate of 3% compounded yearly. The problem involves understanding the present value of an annuity, including one immediate payment followed by 24 annual payments. Ideal for finance students and anyone studying annuities.

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