The following is the only problem that gave me trouble on the homework. The (t+3) and (t+3)/(t) really made me think and I'm assuming that the way I have them is as simplified as they can get. Am I right?
Where value(t) = $1000 (1.07)^t
value(0)= 1000(1.07)^0 =1
value(1)= 1000(1.07)^1= 1070
value(2)= 1000(1.07)^2=1,144,900
Looks like you squared the 1000 as well as the 1.07. Just square the 1.07.
value(t+3)= 1000(1.07)^(t+3)=1070^(t+3)
You don't multiply 1000 by 1.07. The exponentiation precedes the multiplication. So you just leave this as 1000(1.07)^(t+3).
value(t+3)/value(t)= 1000(1.07)^((t+3)/(t))= 1070^((t+3)/(t))
value( (t+3) / t) = 1000(1.07)^((t+3)/(t)), but
value(t+3) / value(t) = 1000(1.07)^(t+3)/(1000 * 1.07^t).
This can be simplified.
Recall that a^(b + c) = a^b * a^c. So
$1000 ( 1.07)^(t + 3) =
$1000 ( 1.07)^t * (1.07)^3.
So
1000(1.07)^(t+3)/(1000 * 1.07^t) =
$1000 ( 1.07)^t * (1.07)^3 / ( $1000 * 1.07^t) =
1.07^3.