If [tex]$10,000 is invested at 14% interest compounded quarterly, find the interest earned in 13 years.

The interest earned in 13 years is $[/tex]_____.

(Do not round until the final answer. Then round to two decimal places as needed.)



Answer :

To find the interest earned for an investment of [tex]$10,000 at an annual interest rate of 14%, compounded quarterly over 13 years, we need to use the compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial sum of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times the interest is compounded per year. - \( t \) is the number of years the money is invested or borrowed for. Given: - \( P = 10,000 \) - \( r = 0.14 \) (14% as a decimal) - \( n = 4 \) (since the interest is compounded quarterly) - \( t = 13 \) First, we need to calculate the accumulated amount \( A \): \[ A = 10,000 \left(1 + \frac{0.14}{4}\right)^{4 \times 13} \] Step 1: Calculate the quarterly interest rate: \[ \frac{0.14}{4} = 0.035 \] Step 2: Calculate the exponent \( 4 \times 13 \): \[ 4 \times 13 = 52 \] Step 3: Add 1 to the quarterly interest rate: \[ 1 + 0.035 = 1.035 \] Step 4: Raise this to the power of 52: \[ 1.035^{52} \] Step 5: Multiply by the principal amount: \[ 10,000 \times 1.035^{52} \] Using these computations, we find: \[ A = 10,000 \left(1.035^{52}\right) \] After calculating the precise value, we subtract the principal amount to find the interest earned: \[ \text{Interest Earned} = A - P \] Thus, \[ \text{Interest Earned} = 10,000 \left(1.035^{52}\right) - 10,000 \] Finally, we get the result: The interest earned in 13 years is \$[/tex]49,827.13 (rounded to two decimal places).