Find the required annual interest rate to the tenth of a percent for $5200 to grow to $6500 if interest is compounded quarterly for five years

It must be invested at 4.5%.

The compound interest formula is

A=p(1+(r/n))^(nt), where A is the total amount including interest earned, p is the principal invested, r is the rate, n is the number of times per year the interest is compounded, and t is the number of years.

With our information we have:

6500=5200(1+(r/4))^(4×5)

6500=5200(1+(r/4))^20

Cancel 5200 by dividing:
6500/5200 = [5200(1+(r/4))^20]/5200
1.25=(1+(r/4))^20

Using a calculator, take the 20th root of each side:

1.25^(1/20) = [(1+(r/4))^20]^(1/20)
1.011219651 = 1+(r/4)

Subtract 1 from each side:
1.011219651-1 = 1+(r/4)-1
0.011219651=r/4

Multiply each side by 4:
0.011219651×4 = (r/4)×4
0.0448786 = r
0.045 = r