Evaluating a geometric series in sigma notation

Key concept: This problem asks you to evaluate a finite geometric series expressed with summation notation. The core skill is identifying the first term, common ratio, and number of terms, then applying the geometric series formula correctly.

Step by step

Recognize the series pattern

The expression is

$\sum_{n=1}^{24} 100(1.27)^n.$

This is a geometric series because each term is obtained by multiplying the previous term by the same common ratio, $1.27$. The sigma notation tells us there are 24 terms, starting at $n=1$ and ending at $n=24$.

The first term is

$a_1=100(1.27)^1=127,$

and the common ratio is

$r=1.27.$

Use the geometric series formula

For a finite geometric series,

$S_n=a_1\frac{1-r^n}{1-r} \quad \text{or} \quad S_n=a_1\frac{r^n-1}{r-1}.$

Since $r>1$, the second form is often easier to use:

$S_{24}=127\cdot\frac{1.27^{24}-1}{1.27-1}.$

Now simplify the denominator:

$1.27-1=0.27.$

So

$S_{24}=127\cdot\frac{1.27^{24}-1}{0.27}.$

Compute and round

At this point, use a calculator to evaluate $1.27^{24}$. The exact value is not expected by hand in most settings. After computing, round the final sum to the nearest integer, as the prompt requests.

Because every term is positive and the ratio is greater than 1, the later terms contribute heavily to the total. That is why a small calculator error in the exponent can affect the final sum, so it is important to keep enough precision during intermediate steps.

Key method to remember

When a series is written with sigma notation, always identify:

• the first term,
• the common ratio,
• the number of terms,
• and the correct geometric series formula.

For this problem, the series is geometric with 24 terms, and the proper setup is the main skill being tested.

Pitfall alert

A common error is to treat $100(1.27)^n$ as if the first term were 100 rather than $100(1.27)^1=127$. Another mistake is forgetting that the summation starts at $n=1$, not $n=0$, which changes the term count and the first term. Students also sometimes use the arithmetic series formula by accident because they see a sum, but the constant multiplier here makes it a geometric series, not an arithmetic one.

Try different conditions

If the same question were changed to $\sum_{n=0}^{24}100(1.27)^n,$ the first term would be 100 instead of 127, and the sum would be larger because it includes one extra initial term. A variant such as $\sum_{n=1}^{24}80(0.95)^n$ would still be geometric, but it would be a decreasing series with ratio less than 1. The solution method is the same, but the numerical behavior changes.

Further reading

common ratio, finite geometric series, sigma notation