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Worked solution

Value annual deposits immediately after the tenth payment

A man deposits $200 at the beginning of each year into an account paying 5%5\% interest compounded annually. How much is in the account immediately after the tenth deposit?

CJC · Sequences and Series · Example 36 (Finance)

Answer

$2515.58 (to the nearest cent).

Compare your reasoning

Tabulate the first few years

YearImmediately after depositEnd of year1200200q2200q+200200q2+200q3200q2+200q+200200q3+200q2+200q4200q3+200q2+200q+200200q4+200q3+200q2+200q\begin{array}{c|l|l}\text{Year}&\text{Immediately after deposit}&\text{End of year}\\\hline1&200&200q\\2&200q+200&200q^2+200q\\3&200q^2+200q+200&200q^3+200q^2+200q\\4&200q^3+200q^2+200q+200&200q^4+200q^3+200q^2+200q\end{array}

Let q=1.05. As the source callout suggests, compare balances at the beginning and end of each year. Interest is applied before the next beginning-of-year deposit.

Stop immediately after the tenth deposit

B10=200+200q+200q2+⋯+200q9\begin{aligned}&B_{10}\\ & =200+200q+200q^2+\cdots\\ &{}+200q^9\end{aligned}

The oldest payment has earned nine years of interest; the newest has earned none. There are ten terms, with exponents zero through nine.

Sum the finite geometric progression

B10=2001.0510−11.05−1=2515.578507…≈2515.58\begin{aligned}&B_{10}\\ & =200\frac{1.05^{10}-1}{1.05-1}\\ & =2515.578507\ldots\approx2515.58\end{aligned}

Factor out 200 and use first term 1, ratio 1.05 and ten terms. Round only the final balance.

How did this question go?

Common mistakes

Immediately after the tenth deposit is the beginning of year ten. Applying another year of interest changes the requested endpoint.