Question

Suppose $1000 deposited for six years in an account paying 5% interest compounded annually.
a) Find the final amount in the account
b) Find the actual amount of interest earned.
Answer clearly by showing your work and step by step

Answer 1

To find the final amount in the account and the actual amount of interest earned, we can use the formula for compound interest:

\[
A = P(1 + r)^n
\]

where:
- \(A\) is the final amount in the account after \(n\) years,
- \(P\) is the principal amount (the initial deposit),
- \(r\) is the annual interest rate (in decimal form),
- \(n\) is the number of years the money is invested or borrowed.

### Given:
- \(P = 1000\)
- \(r = 5\% = 0.05\)
- \(n = 6\)

### Step a: Find the final amount in the account

Plugging in the values:

\[
A = 1000(1 + 0.05)^6
\]

Calculating \(1 + 0.05\):

\[
1 + 0.05 = 1.05
\]

Now calculating \(1.05^6\):

\[
1.05^6 \approx 1.34009564
\]

Now, multiply this by \(1000\):

\[
A = 1000 \times 1.34009564 \approx 1340.10
\]

Thus, the final amount in the account after 6 years is approximately:

\[
A \approx 1340.10
\]

### Step b: Find the actual amount of interest earned

The interest earned can be calculated as:

\[
\text{Interest} = A - P
\]

Substituting the values we have:

\[
\text{Interest} = 1340.10 - 1000
\]

Calculating the interest:

\[
\text{Interest} = 340.10
\]

### Summary of Results:

a) The final amount in the account after 6 years is approximately **$1340.10**.
b) The actual amount of interest earned is approximately **$340.10**.