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Ron sample is the grand prize winner in a college tuition essay contest
awarded through a local organization’s scholarship fund. The winner
receives at the beginning of each year for the next 3 years.
How much (in ) should be invested at interest compounded annually t.
laward the prize? (Round your answer to the nearest cent.)

Ask by Garza Malone. in the United States

Mar 21,2025

Question

File Edit format View Help
Ron sample is the grand prize winner in a college tuition essay contest
awarded through a local organization’s scholarship fund. The winner
receives at the beginning of each year for the next 3 years.
How much (in ) should be invested at interest compounded annually t.
laward the prize? (Round your answer to the nearest cent.)

Ask by Garza Malone. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

The prize is an annuity due because the payments are made at the beginning of each year. The three payments occur at times $$ t = 0 $$, $$ t = 1 $$, and $$ t = 2 $$. The present value (PV) for an annuity due is calculated by summing the discounted value of each payment.

1. The first payment is made at $$ t = 0 $$ (no discounting):
   $$
   \text{Payment at } t=0 = \$9{,}000
   $$

2. The second payment is at $$ t = 1 $$ and is discounted one year:
   $$
   \text{Payment at } t=1 = \frac{\$9{,}000}{1.06}
   $$

3. The third payment is at $$ t = 2 $$ and is discounted two years:
   $$
   \text{Payment at } t=2 = \frac{\$9{,}000}{1.06^2}
   $$

The total present value is:
$$
\text{PV} = 9000 + \frac{9000}{1.06} + \frac{9000}{1.06^2}
$$

Calculating each term:

- For $$ t=1 $$:
  $$
  \frac{9000}{1.06} \approx 9000 \times 0.943396 = 8490.56
  $$

- For $$ t=2 $$:
  $$
  \frac{9000}{1.06^2} \approx 9000 \times 0.889996 = 8009.97
  $$

Summing these:
$$
\text{PV} \approx 9000 + 8490.56 + 8009.97 \approx 254, \; \text{(sum step-by-step: }9000 + 8490.56 = 17490.56; \quad 17490.56+8009.97=25500.53\text{)}
$$

Thus, the amount that should be invested is approximately:
$$
\boxed{\$25{,}500.53}
$$
Approximately \$25,500.53 should be invested.

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