Question

Compound interest is interest earned on both the initial investment and the previously earned interest. An initial amount of money, the principal P 0, is invested in an account that pays an annual interest rate r (written as a decimal), compounded n times per year. The amount in the account at the end of each time period is the principal plus the interest.
For example, $1,000 is invested at a 2% interest rate and compounded quarterly. Then P 0 = 1,000, r = 0.02, and n = 4.

Describe the location of the vertex of the parabola relative to the x-axis. How many zeros does the polynomial have?

(Fill in the blanks)

Time period, Amount in the account (with rounding)
Initial principal, P 1,000
End of the first period 1,000 + 1,000 + 0.02/4 = _____

End of the second period ____ + ____ * 0.02/4 = _____

End of the third period ____ + ____ * 0.02/4 = _____

End of the fourth period ____ + ____ * 0.02/4 = _____

Answer 1

The amount P as a function of t (in years) is given by

P(t) = P0 (1 + r/n)^(t n)

So if n = 4, and r = 0.02, and P0 = 1000, then

P(t) = 1000 (1 + 0.02/4)^(4 t) = 1000 (1 + 0.005)^(4 t)

At the end of the first quarter, t = 1/4, so

P(1/4) = $1000 (1.005)^(1) = $1005

At the end of the second quarter, t = 1/2 , therefore

P(1/2) = $1000 (1.005)^(2) = $1000 (1.010025) = $1010.03

At the end of the third quarter , t = 3/4, therefore

P(3/4) = $1000 (1.005)^(3) = $1000 (1.015075125) = $1015.08

At the end of the year, t = 4, therefore

P(1) = $1000 (1.005)^4 = $1000 (1.020150500625) = $1020.15

As for the second question, after the first period (quarter),
the formula becomes

P = P0 (1.005)^1 = 1.005 P0

which is choice A.