Question
To save money for his daughter's college tuition, Shen invests $258 every quarter in an annuity that pays 6% interest, compounded quarterly. Payments will be made at the end of each quarter. Find the total value of the annuity in 21 years.
Answers
GPT 3.5
We can solve this problem using the formula for the future value of an annuity:
\[A = P \times \left( \dfrac{(1 + r)^n - 1}{r} \right)\]
Where:
A = Total value of the annuity
P = Amount invested at the end of each payment period
r = Interest rate per period
n = Number of payment periods
In this case,
P = $258
r = 6% per year = 6/100 = 0.06 per quarter
n = 21 years * 4 quarters = 84 quarters
Let's substitute these values into the formula and calculate the total value of the annuity:
\[ A = 258 \times \left( \dfrac{(1 + 0.06)^{84} - 1}{0.06} \right) \]
Using a calculator, we find that
\[ A \approx \$27,855.76 \]
Therefore, the total value of the annuity in 21 years will be approximately $27,855.76.
\[A = P \times \left( \dfrac{(1 + r)^n - 1}{r} \right)\]
Where:
A = Total value of the annuity
P = Amount invested at the end of each payment period
r = Interest rate per period
n = Number of payment periods
In this case,
P = $258
r = 6% per year = 6/100 = 0.06 per quarter
n = 21 years * 4 quarters = 84 quarters
Let's substitute these values into the formula and calculate the total value of the annuity:
\[ A = 258 \times \left( \dfrac{(1 + 0.06)^{84} - 1}{0.06} \right) \]
Using a calculator, we find that
\[ A \approx \$27,855.76 \]
Therefore, the total value of the annuity in 21 years will be approximately $27,855.76.