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Let $A_t$ be the amount of money at the beginning of year $t$ and let $r$ be the interest rate. Assume that each year, we invest an additional amount of $Q$ Then we can recursively write this as: $$ A_t = (1+r)A_{t-1} + Q. $$ So the amount at time $t$, $A_t$ is the amount last year $A_t$ plus a percentage $r$ of $A_t$ plus $Q$. We would like to find a closed ...


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Let $a =$ Principle Let $b = 1$ + interest rate Let $c_{1} =$ future value after year $1$ Let $d =$ additional investment $c_{1} = ab$ $c_{2} = (c_{1} + d)b$ $c_{3} = (c_{2} + d)b$ $c_{4} = (c_{3} + d)b$ $c_{5} = (c_{4} + d)b$ simplified form for $c_{2}$ $c_{2} = (c_{1} + d)b= c_{1}b + db$ simplified form for $c_{3}$ $c_{3} = (c_{2} + d)b = (c_{1}b + db + d)...


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