Final answer:
Nicholas would need an annual interest rate of approximately 1.82% for his $13,000 investment to grow to $14,300 in 4 years with quarterly compounding, using the compound interest formula A = P(1 + r/n)^(nt).
Step-by-step explanation:
Nicholas wants to know the annual interest rate required for his $13,000 investment to grow to $13,000 + $1,300 = $14,300 in 4 years with quarterly compounding. The compound interest formula is A = P(1 + r/n)nt, where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.
First, we will isolate r in the formula and solve for it using the given values:
- P = $13,000
- A = $14,300
- n = 4 (since the interest is compounded quarterly)
- t = 4 years
Now substitute the values into the formula and solve for r:
$14,300 = $13,000(1 + r/4)4 × 4
1.1 = (1 + r/4)16
Take the 16th root of both sides:
(1 + r/4) = 1.11/16
Now calculate r:
r = 4 × (1.11/16 - 1)
r ≈ 0.0182 or 1.82%
Nicholas would need an annual interest rate of approximately 1.82% for his investment to grow to $14,300 in 4 years with quarterly compounding.