To determine the annual nominal compounding interest rate $r$, we can use the future value formula for an ordinary annuity:

$FV = P \times \frac{(1 + r)^n - 1}{r}$

Where:

• $FV$ is the future value of the annuity (in this case, $5701.60),
• $P$ is the annual payment (in this case, $1000),
• $r$ is the annual nominal interest rate (which we want to find),
• $n$ is the number of years (in this case, 5 years).

We are given:

• $FV = 5701.60$,
• $P = 1000$,
• $n = 5$.

Thus, the equation becomes:

$5701.60 = 1000 \times \frac{(1 + r)^5 - 1}{r}$

To solve for $r$, we can either use graphical approximation techniques or an equation solver.

Let me calculate the value of $r$.The annual nominal compounding interest rate that the annuity has earned is approximately 6.57%.

Would you like further details or explanations? Here are some related questions to expand on this concept:

1. How does changing the number of years affect the annuity's future value?
2. What if the payments were made at the beginning of each year (an annuity due)?
3. How would this formula change if the compounding was monthly instead of annually?
4. What happens to the interest rate if the future value were higher?
5. How can you manually calculate interest rates for more complex annuities?

Tip: When solving annuity problems, always ensure that the payment period and compounding frequency align.