45 Points!!!!!!!!!!!!!!Alexander deposited money into his retirement account that is compounded annually at an interest rate of 7%. Alexander thought the equivalent quarterly interest rate would be 2%. Is Alexander correct? If he is, explain why. If he is not correct, state what the equivalent quarterly interest rate is and show how you got your answer.

The equation for compound interest is:

[tex]A = P(1+ \frac{r}{n})^{ nt} [/tex]

Where r is the interest rate and n is the number of times per year it's applied.
 
Annually n = 1 and 7% interest r = 0.07
The quarterly rate 2% is already quartered 0.02 = r/n .

[tex](1+0.07)= (1+0.02 ) ^{4} \\ \\ 1.07 = (1.02) ^{4} \\ \\ 1.07 \neq 1.082 [/tex]

You can see that Alexander is incorrect. A quarterly compound interest rate of 2% will accrue more interest than a 7% compound annual interest rate.

[tex] (1+0.07) = (1+ r) ^{4} \\ \\ 1.07 = (1+r) ^{4} \\ \\ \sqrt[4]{1.07} = r \\ \\ \sqrt[4]{1.07} - 1 = r \\ \\ r = 0.017 [/tex]

1.7% compound quarterly