High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize High School Algebra II Textbook (Common Core) > Sequences & Series

Geometric Series

Geometric SequencesPrevious Section
Pascal's Triangle and the Binomial ExpansionNext Section
Popular Courses
Find My Course
0:00 / 0:00

Geometric Series

A geometric series is the sum of the terms of a geometric sequence.

Wize Concept
Recall, for a geometric sequence:
  • aa is the first term (also called t1t_1)
  • rr is the common ratio (found by dividing any term by the term that comes before it)

The sum of the first nn terms of a geometric series (called the partial sum) is given by:

Sn=a(rn−1)r−1,r≠1\boxed{\quad S_n = \dfrac{a(r^n-1)}{r-1}, \quad r\ne 1 \quad}

Note: an equivalent formula can be written using the first term, t1t_1, and the (n+1)(n+1)th term, tn+1t_{n+1}:

Sn=tn+1−t1r−1,r≠1\boxed{\quad S_n = \dfrac{t_{n+1}-t_1}{r-1}, \quad r\ne 1 \quad}
PAGE BREAK
Example

Consider the geometric sequence 2,6,18,54, …2,6,18,54,\ \dots

What is the associated geometric series?
2+6+18+54+...

What is the common ratio? r=r=
3

What is the sum of the first 4 terms of this series?

The first term is a=2a=2, and we want to find the sum of the first n=4n=4 terms:
Sn=a(rn−1)r−1S4=2(34−1)3−1=2(81−1)2=80\begin{aligned} S_n &= \dfrac{a(r^n-1)}{r-1}\\[1em] S_4 &= \dfrac{2(3^4-1)}{3-1} = \dfrac{2(81-1)}{2} = \boxed{80} \end{aligned}

When nn is small, we can easily check: 2+6+18+54=80✓2+6+18+54 = 80 \quad \colorThree{\checkmark}

How to derive the Geometric Series Formula


The sum of the first nn terms of a geometric series with rr as the common ratio is given by the formula:

a(rn−1)r−1\displaystyle \frac{a(r^n - 1)}{r-1}

Let's see how we can build this formula so we can better understand where it comes from.

1. We can begin with the expression for the first nn terms of a geometric series.

a+ar+ar2+⋯+arn−1a + ar + ar^2 + \cdots + ar^{n-1}
Note that since our first term is just aa, this is the same as ar0ar^0. So to have exactly nn terms, the last term must be arn−1ar^{n-1}.

2. All of these have a common factor of aa. Factoring this out we have

a(1+r+r2+⋯+rn−1)a(1 + r + r^2 + \cdots + r^{n-1})
3. In parenthesis, we have a polynomial. This can be replaced using 1+r+r2+⋯+rn−1=1−rn1−r1 + r + r^2 + \cdots + r^{n-1} = \frac{1 - r^n}{1 - r} . One way to verify this step is by performing long division.

a(1−rn1−r)a\left(\displaystyle\frac{1-r^n}{1-r} \right)

4. We can reverse the order of subtraction in the top and the bottom. This is the same as factoring out a -1

a((−1)rn−1(−1)(r−1))a\left(\displaystyle\frac{(-1)r^n -1}{(-1)(r-1)} \right)
5. Finally we cancel out the -1's and arrive at the formula for the sum of a geometric series.

a(rn−1r−1)a\left(\displaystyle\frac{r^n - 1}{r - 1} \right)

These steps show us that that the following are equal:

a+ar+ar2+⋯+arn−1=a(rn−1r−1)a + ar + ar^2 + \cdots + ar^{n-1} = a\left(\displaystyle\frac{r^n - 1}{r - 1} \right)

0:00 / 0:00

Example: Geometric Series

Would you rather:
  1. receive one million dollars ($1 000 000) today, or
  2. receive a penny ($0.01) today, and then double that tomorrow, and double again the next day, and so on for 30 days?
Let's use geometric series to calculate the total amount of money received in the second scenario, where n=30n=30 days.

In this scenario, the geometric series is: 0.01+0.02+0.04+0.08+0.16+⋯+t300.01+0.02+0.04+0.08+0.16+\dots+t_{30}

The first term of the series corresponds to the amount on the first day: t1=a=0.01t_1 =a =0.01

The common ratio is how much we multiply the amount by every day: r=2r=2

Sn=a(rn−1)r−1S30=0.01(230−1)2−1=0.01(230−1)Note: 230=1 073 741 824=$ 10 737 418.23\begin{aligned} S_n &= \dfrac{a(r^n-1)}{r-1}\\[1em] S_{30} &= \dfrac{0.01(2^{30}-1)}{2-1}\\[1em] &= 0.01(2^{30}-1) \quad \text{Note: }2^{30} = 1\,073\,741\,824 \\[0.5em] &= \$\,10\,737\,418.23 \end{aligned}

Therefore, by waiting 30 days, the second scenario allows you to receive more than 10 times what is offered in the first scenario!

This is the power of geometric growth, which is the same exponential growth but with discrete steps (not continuous).

Practice: Geometric Series

Select all of the options that are geometric series.

Practice: Geometric Series

a) Determine the sum of the geometric series 1600+800+400+⋯+251600+800+400+\dots+25.

b) Determine the number of terms in the geometric series −2+6−18+⋯+tn-2+6-18+\dots+t_n whose sum is 3280.

Practice: Geometric Series


A diver wants to practice holding her breath once per day. On her first day of training, she holds her breath for 30 seconds. Every day, she manages to hold her breath 10% longer than the day before.

Over these two weeks, what was the average amount of time she managed to hold her breath every day?