Find the amount of interest in the third year if $5000 is invested for 6 years at 8% p.a. compound interest.

To find the amount of interest in the third year on an investment of $5000 for 6 years at an 8% per annum compound interest rate, we need to calculate the total amount at the end of the third year and the end of the second year, then find the difference between these amounts. 

The formula for the compound interest is:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

Here:
- $ P = 5000 $
- $ r = 0.08 $
- $ n = 1 $ (compounded annually)
- $ t $ is the number of years

First, calculate the amount after 3 years:

\[ A_3 = 5000 \left(1 + 0.08\right)^3 \]
\[ A_3 = 5000 (1.08)^3 \]

Now, calculate $ (1.08)^3 $:

\[ (1.08)^3 \approx 1.2597 \]

Then,

\[ A_3 \approx 5000 \times 1.2597 \]
\[ A_3 \approx 6298.5 \]

Next, calculate the amount after 2 years:

\[ A_2 = 5000 \left(1 + 0.08\right)^2 \]
\[ A_2 = 5000 (1.08)^2 \]

Now, calculate $ (1.08)^2 $:

\[ (1.08)^2 \approx 1.1664 \]

Then,

\[ A_2 \approx 5000 \times 1.1664 \]
\[ A_2 \approx 5832 \]

The interest earned in the third year is the difference between the amount after the third year and the amount after the second year:

\[ \text{Interest in 3rd year} = A_3 - A_2 \]
\[ \text{Interest in 3rd year} \approx 6298.5 - 5832 \]
\[ \text{Interest in 3rd year} \approx 466.5 \]

So, the amount of interest earned in the third year is approximately $466.50.

Would you like details or have any questions? Here are some related questions to explore further:

1. How much total interest is earned at the end of 6 years?
2. What would the interest be in the fourth year?
3. How would the interest in the third year change if the rate was 10% instead of 8%?
4. What is the formula to calculate the interest for any specific year in compound interest?
5. How much interest is earned in the first three years combined?
6. What happens to the interest earned if the investment period is extended to 10 years?
7. How does the frequency of compounding affect the interest earned in a specific year?
8. What is the total amount after 6 years?

**Tip:** To find the interest earned in a specific year for compound interest, calculate the total amount at the end of that year and subtract the total amount at the end of the previous year.

Learn how to calculate compound interest for an investment of $5000 at an 8% per annum rate over 6 years. Find the amount of interest earned in the third year using step-by-step calculations and the compound interest formula. Suitable for high school students and anyone interested in financial mathematics.

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