A savings account pays a 3% nominal annual interest rate and has a balance of $1,000. Any interest earned is

deposited into the account and no further deposits or

withdrawals are made.

Write an expression that represents the balance in one

year if interest is compounded annually.

The compound interest formula is given by,

A\ =\ P\left(\ 1+\frac{r}{n}\right)^{nt}

where P is the initial amount which is $1000, the rate of interest is 0.03, n = 1 (since the interest is compounded once a year, annually), t = 1 year,

Plug in these find A=1000(1+0.03)\ =\ 1030

So, the balance in one year, of compounded annually is $1030

This is the case of compound interest since the interest is deposited back into the and no further deposits or withdrawals are made.

P(1) = Principal after 1 year = P(0) + interest after one year = P(0) + r P(0) = (1 + r) P(0)

P(2) = Principal after 2 years = P(1) + interest after second year = P(1) + r P(1) = (1 + r) P(1) = (1 + r)^2 P(0).

So, the amount after t years = 1000 (1 + 0.03)^t = 1000 x 1.03^t.