- I plan to save $100 at the start of each month for the next two years. How much will I have at the end of the two-year period (to the nearest cent) if interest rates remain at 8% per annum (compounded monthly)?
I am having trouble answering this question confidently. A walkthrough would be amazing.
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This is an annuity. You pay at the start of each month, so it is an 'annuity due'. You want the future value (how much will it be worth at the end?). You may have formulas from lectures that give you the answer. If not, note that Ross's answer is a geometric series $$ R a \left( 1 + a + a^2 + \cdots + a^{23} \right) $$ What is the sum of a geometric series?
First, $8\%$ per year is $\frac 23\%$ per month. The first $\$100$ you save is on deposit for $24$ months. Each month it is multiplied by $1+\frac 23\%=\frac {302}{300}$. So at the end it is worth $100 \left( \frac {302}{300} \right) ^{24}$. Each successive deposit is there for one less month. This gives you a geometric progression to sum: $$100\left(( \frac {302}{300})+( \frac {302}{300})^2+( \frac {302}{300})^3 \dots ( \frac {302}{300})^{24}\right)$$