01 / 21

Geometric Series
Formula

Derive and apply the formula for the sum of a finite geometric series, and use it to solve real-world problems like mortgage payments and investment growth.

Algebra 2

What Is a Geometric Sequence?

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed value called the common ratio \( r \).

3 6 12 24 48 ×2 ×2 ×2 ×2
\(a_1\)
\(a_2\)
\(a_3\)
\(a_4\)
\(a_5\)

Key Definitions

First Term \( a_1 \)
The starting value of the sequence. Sometimes written simply as \( a \).
Common Ratio \( r \)
The constant multiplier between consecutive terms: \( r = \frac{a_{n+1}}{a_n} \).
Number of Terms \( n \)
How many terms are being added together in the series.

Why Does This Matter?

Geometric series appear everywhere: compound interest, mortgage payments, population growth, and even computer science algorithms. Instead of adding dozens of terms one by one, the formula gives us the answer in a single calculation.

Finance
Monthly mortgage or annuity payments are sums of geometric series.
Science
Radioactive decay and drug dosage models follow geometric patterns.
Tech
Algorithm analysis often involves summing powers of 2.

Deriving the Formula

Start by writing the sum and then multiply both sides by \( r \):

Notice that nearly every term on the right side of the two equations is identical. That is the key insight.

The Subtraction Trick

The Geometric Series Formula

$$ S_n = \frac{a_1(1 - r^n)}{1 - r}, \quad r \neq 1 $$
\(S_n\) = sum
\(a_1\) = first term
\(r\) = common ratio
\(n\) = # of terms

Arithmetic vs Geometric

Arithmetic Series Geometric Series
Pattern Add a constant \( d \) Multiply by a constant \( r \)
Example 2, 5, 8, 11, ... 2, 6, 18, 54, ...
Sum Formula \( S_n = \frac{n}{2}(a_1 + a_n) \) \( S_n = \frac{a_1(1 - r^n)}{1 - r} \)
Growth Linear Exponential

How to Apply the Formula

1
Identify \( a_1 \): Find the first term of the series.
2
Find \( r \): Divide any term by the one before it: \( r = \frac{a_2}{a_1} \).
3
Count \( n \): Determine the number of terms being summed.
4
Plug in: Substitute into \( S_n = \frac{a_1(1 - r^n)}{1 - r} \).
5
Simplify: Compute \( r^n \) first, then the numerator, then divide.

Example 1: Basic Series

Find the sum of the first 6 terms of the series: \( 5 + 10 + 20 + 40 + \cdots \)

1
Identify: \( a_1 = 5 \), \( r = \frac{10}{5} = 2 \), \( n = 6 \).
2
Substitute: \( S_6 = \frac{5(1 - 2^6)}{1 - 2} \).
3
Compute: \( 2^6 = 64 \), so \( S_6 = \frac{5(1 - 64)}{-1} = \frac{5(-63)}{-1} = 315 \).

Example 2: Fractional Ratio

Find the sum: \( 81 + 27 + 9 + 3 + 1 \).

1
Identify: \( a_1 = 81 \), \( r = \frac{27}{81} = \frac{1}{3} \), \( n = 5 \).
2
Substitute: \( S_5 = \frac{81\!\left(1 - \left(\frac{1}{3}\right)^5\right)}{1 - \frac{1}{3}} \).
3
Compute: \( \left(\frac{1}{3}\right)^5 = \frac{1}{243} \). So, \( S_5 = \frac{81\left(1 - \frac{1}{243}\right)}{\frac{2}{3}} = \frac{81\left(\frac{242}{243}\right)}{\frac{2}{3}} = \frac{\frac{242}{3}}{\frac{2}{3}} = 121 \).

Example 3: Sigma Notation

Evaluate: \( \displaystyle\sum_{k=1}^{8} 3 \cdot 2^{k-1} \).

1
Read the sigma: First term when \( k = 1 \): \( 3 \cdot 2^0 = 3 \). So \( a_1 = 3 \), \( r = 2 \), \( n = 8 \).
2
Substitute: \( S_8 = \frac{3(1 - 2^8)}{1 - 2} = \frac{3(1 - 256)}{-1} \).
3
Simplify: \( S_8 = \frac{3(-255)}{-1} = 765 \).

Real World: Savings Growth

You deposit $200 at the start of each year into an account earning 5% annual interest. How much do you have after 10 years?

Visualizing Partial Sums

For the series \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \), watch how the partial sums approach \( 2 \):

0 0.5 1.0 1.5 2.0 2 1
\(S_1\)
 1.5
\(S_2\)
 1.75
\(S_3\)
 1.88
\(S_4\)
 1.94
\(S_5\)
 1.97
\(S_6\)
 1.98
\(S_7\)

Common Mistakes

✕ Wrong Exponent
Using \( r^n \) instead of \( r^{n-1} \) for the last term, or miscounting the number of terms \( n \).
✕ Confusing Formulas
Swapping the arithmetic sum formula \( \frac{n}{2}(a_1 + a_n) \) with the geometric formula. Check the pattern first.
✕ Sign Errors
Forgetting that \( 1 - r \) is negative when \( r > 1 \). Two negatives in the fraction cancel to give a positive sum.

Summary Cheat Sheet

Concept Formula / Rule
\( n \)-th term \( a_n = a_1 \cdot r^{n-1} \)
Finite sum (\( r \neq 1 \)) \( S_n = \frac{a_1(1 - r^n)}{1 - r} \)
Infinite sum (\( |r| < 1 \)) \( S_\infty = \frac{a_1}{1 - r} \)
Common ratio \( r = \frac{a_{n+1}}{a_n} \)
Restriction \( r \neq 1 \) for the finite formula

Question 1 of 5

Find the sum of the first 5 terms of the geometric series where \( a_1 = 4 \) and \( r = 3 \).

Question 2 of 5

What is the common ratio of the series \( 100 + 50 + 25 + 12.5 + \cdots \)?

Question 3 of 5

Evaluate: \( \displaystyle\sum_{k=1}^{6} 2 \cdot 5^{k-1} \).

Question 4 of 5

A ball is dropped from 16 feet. Each bounce reaches \( \frac{3}{4} \) of the previous height. What is the total distance the ball falls during the first 4 drops (not counting bounces up)?

Question 5 of 5

In the derivation of the geometric series formula, what is the purpose of multiplying \( S_n \) by \( r \)?