This is from Discrete Mathematics and its Applications
Definition of recurrence relation from book
From my understanding, compounded annually means that every year(annually) the account will increase by that rate.
So for 18a, to relate the current year to the previous year(recurrence relation definition) I got $b_n = 1.09b_{n-1}$ or $1.0b_{n-1} + 0.09b_{n-1}$ to show that the account increases by that 9% rate every year.
18b. I got the equation $1000(1.09)^t$ where $t$ is the number of years passed.
18c. I used my equation from 18b and got $1000(1.09)^{100}$ or \$5529040.792.
Is all of that correct? Is there a better way i could explain things?
Your answers are mainly correct but would be easier to read if you used LaTeX
$b_n=1.09b_{n-1}$ is a better answer than $b_n=1.0b_{n-1}+0.09b_{n-1}$
$b_n =1000 \times (1.09)^n$ might be better than introducing $t$
$\$1000 \times (1.09)^{100} \approx \$5529040.79$ but you have an extra $5$ (a typo?)