Geometric Series

How to find the sum of a geometric sequence in one calculation.

In 2020, a single post on social media asked people to share it with two friends, each of whom would share it with two more, and so on. After 10 rounds of sharing, the post had reached over a thousand accounts. If you wanted to know the total number of shares at any point — not just the last round, but every round added together — you needed more than a geometric sequence. You needed a geometric series.

A geometric sequence lists terms. A geometric series adds them. If a geometric sequence has first term $a_1$ and common ratio $r$ , the series is the sum of its first $n$ terms:

S_n = a_1 + a_1 r + a_1 r^2 + a_1 r^3 + \cdots + a_1 r^

Writing out every term gets tedious fast. There is a closed formula. Multiply both sides of that sum by $r$ , and you get $rS_n = a_1 r + a_1 r^2 + \cdots + a_1 r^n$ . Subtract the second equation from the first and almost everything cancels — only the first term of $S_n$ and the last term of $rS_n$ survive. That gives $S_n - rS_n = a_1 - a_1 r^n$ , which factors into the formula every Algebra II student needs:

S_n = \frac,, \quad r \neq 1

Here $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. The restriction $r \neq 1$ matters: if every term is identical, the formula breaks down because you'd divide by zero. In that case the sum is just $n \cdot a_1$ .

Sigma notation gives a compact way to write a series. The symbol $\Sigma$ means "sum the following expression for each integer value of the index." The expression

\sum_,^, a_1 r^

says: start with $k = 1$ , plug each integer up through $k = n$ into $a_1 r^$ , and add the results. When $k=1$ the exponent is zero, so the first term is $a_1 r^0 = a_1$ . When $k=2$ you get $a_1 r^1$ , and so on. This matches the series written out above.

The formula works for any value of $r$ except 1. When $|r| < 1$ , each term is smaller than the last and the partial sums settle toward a finite number. When $|r| > 1$ , the terms grow and the sums grow with them. Both cases use the same formula.

Here is the social media example computed directly.

Total shares after 10 rounds, starting with 1 post shared to 2 each time

a_1 = 1, \quad r = 2, \quad n = 10

The chain starts with 1 share, doubles each round, and runs for 10 rounds.

S_{10} = \frac{1(1 - 2^{10})}{1 - 2}

Plug straight into the formula. Keep the subtraction order exactly as written — the signs are easy to mix up.

S_{10} = \frac{1(1 - 1024)}{-1}

2 to the 10th power is 1024.

S_{10} = \frac{-1023}{-1}

Negative divided by negative.

S_{10} = 1023 \checkmark

Over a thousand total shares from just 10 rounds of doubling.

Now a problem where the ratio is a fraction, and it is given using sigma notation.

Evaluating a sigma notation geometric series

\sum_{k=1}^{5} 48 \left(\frac{1}{2}\right)^{k-1}

Read the sigma: the first term is 48 times (1/2) to the zero power, which is 48. The ratio is 1/2. There are 5 terms.

a_1 = 48, \quad r = \frac{1}{2}, \quad n = 5

Pull out the three pieces the formula needs before you plug anything in.

S_5 = \frac{48\!\left(1 - \left(\frac{1}{2}\right)^5\right)}{1 - \frac{1}{2}}

Substitute into the formula.

S_5 = \frac{48\!\left(1 - \frac{1}{32}\right)}{\frac{1}{2}}

(1/2) to the 5th power is 1/32.

S_5 = \frac{48 \cdot \frac{31}{32}}{\frac{1}{2}}

1 minus 1/32 equals 31/32.

S_5 = 48 \cdot \frac{31}{32} \cdot 2

Dividing by 1/2 is the same as multiplying by 2.

S_5 = \frac{48 \cdot 62}{32} = \frac{2976}{32} = 93 \checkmark

Multiply across and simplify. The sum of five terms of this sequence is exactly 93.

Interactive graph — scroll to zoom, drag to pan

The red curve shows the individual terms of the sequence — each one is half the previous. The blue line marks the sum $S_5 = 93$ . Notice how the terms shrink quickly; the fifth term contributes very little compared to the first.

Practice Questions

S_6 \text{ for } a_1 = 3, \; r = 2

\sum_{k=1}^{4} 5 \cdot 3^{k-1}

\sum_{k=1}^{6} 64 \left(-\frac{1}{2}\right)^{k-1}

Regents Corner

On Part II and Part III of the Algebra II Regents, geometric series problems often ask you to identify $a_1$ , $r$ , and $n$ from sigma notation before computing the sum. Show that identification work explicitly — a grader reading your paper needs to see where your numbers came from. Writing the formula with values substituted before simplifying is worth method credit even if arithmetic errors appear later.

When the sigma expression is written as something like 48(1/2)^k instead of 48(1/2)^(k-1), the first term is not 48 — it is 48 times 1/2, which is 24. Students who assume the lower bound k=1 always gives a_1 equal to the coefficient in front of the ratio will get the wrong first term and a wrong answer from the start. Always evaluate the expression at the lower index to find the true first term.

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Arithmetic Series and Sigma Notation

Recursive and Explicit Formulas