So, the question itself is:
On the day of her birth, 1st January 1998, Mary's grandparents invested $x$ dollars in a savings account. They continued to deposit $x$ dollars on the first day of each month thereafter. The account paid a fixed rate of $0.4\%$ interest per month. The interest was calculated on the last day of each month and added to the account.
Let $A(n)$ be the amount in Mary's account in dollars on the last day of the $n$-th month, immediately after the interest had been added.
(a) Find an expression for $A(1)$ and show that $A(2) = 1.004^2x + 1.004x $
At part (a), the answer would be:
$A(1) = 1.004x $
$A(2) = 1.004(A(1) + x) $
$A(2) = 1.004(1.004x + x) $
$A(2) = 1.004^2x + 1.004x $
However, what I can't understand is why we multiply by $1.004$ when finding $A(2)$. Shouldn't we be multiplying by the interest, which is $0.004$?
Best Answer
$A(2)$ is the amount at the end of the second month. A deposit of $x$ may made at the beginning of the first month, so after two months it has grown to $1.004^2x$. A deposit of $x$ was also made at the beginning of the second month. At the end of the second month it has grown to $1.004x.$ So $$A(2)=1.004^2x+1.004x$$
Does it make sense now?