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In the case where the terms of a sequence increase by a constant ratio, the sum of the sequence can be modeled by geometric series. This lesson will introduce how to find the sum of geometric series for both finite and infinite cases.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Finding the Sum

Consider the following sum of a sequence.
a Is the sum finite or infinite?
b If the sum is finite, what is its value? Round the answer to two decimals.
Discussion

Geometric Series

A geometric series is the sum of the terms of a geometric sequence.
The explicit rule of the geometric sequence above is This rule can be used to write the series using sigma notation.
The sum of an infinite or a finite geometric series can be found by using its corresponding formula.
Pop Quiz

Identifying Geometric Series

The following applet shows the first five terms of a sum. Identify whether the given sum is a geometric series or not.

Applet showing some series
Discussion

Finite Geometric Series

Let and be the first term and the common ratio, respectively, of a geometric sequence with terms, where The sum of the related finite geometric series can be found by using the following formula.


Proof

Finite Sum
To derive the formula for a geometric series, a geometric sequence of terms with the first term and the common ratio will be considered.
All the terms will be added to express the series and will be factored out.
Then, both sides of the equation will be multiplied by and the resulting equation simplified.
Notice that the like terms are ordered in minus-plus pairs. This means that after simplifying, they will cancel out and only the first and last terms will remain.
The formula for the sum of a finite geometric series has been derived.

Example

The Number of Cases of Sickness at School

Tearrik goes to North High School. One day he suddenly started feeling sick at school. Later it was determined that according to school records, Tearrik's three best friends started feeling sick the next day, nine more students stayed home sick on the third day, and so on.
spreading the virus
Within a few days, it was discovered that a new kind of viral disease was spreading rapidly through the school, with each infected student transmitting the virus to three other students each day.
a Write an explicit rule to represent the number of students who become infected on a given day.
b Calculate the total number of infected students that there would be in days if the school does not suspend classes.

Hint

a The number of newly-infected students each day makes a geometric sequence.
b Use the formula for the sum of a geometric series.

Solution

a It is given that the number of newly-infected students is triple the number of students that were infected the day before.
number of infected students
This means that a common ratio exists between the number of newly-infected students each day. Therefore, their sum represents a geometric series. With this in mind, an explicit rule can be written to find the number of newly-infected students on the day.
Since only Tearrik fell ill on the first day, the first term of the sequence is and the common ratio Substitute these values into the formula to find the explicit rule for this sequence.
The number of newly-infected students on the day can be represented by the formula
b To calculate the total number of infected students on any given day, the formula for the sum of a finite geometric series will be used.
Since the total number of infected students on the day is required, will be substituted into the formula. Remember that the first term is and the common ratio is
Evaluate right-hand side
If the school does not suspend classes, there will be infected students in days.
Example

Geometric Series in Sigma Notation

As the number of infected students increased, the school decided to start a quarantine period. During the quarantine period at North High School, lessons started being taught online. In one of his math lessons, Tearrik's math teacher introduced a geometric series an example written using sigma notation.

Two example of series shown in the computer screen. Series 1: \sum \limits_{i=1}^{12} 6 \t 3^{i} Series 2: \sum \limits_{n=1}^{7} \t 3^{i}
Calculate the given sum.

Hint

Use the formula for the sum of a finite geometric series.

Solution

The given sum is a finite geometric series. Therefore, the formula for the sum of a finite geometric series can be used to find this sum.
In this formula, is the first term, is the common ratio, and is the number of terms. First, substitute into the expression given on the right side of the sigma notation to find
The first term is Notice that the summation index is placed as a single exponent of meaning that the next term can be found by multiplying the previous one by This makes the common ratio Another way to find is to divide the second term by the first. Start by substituting into the given expression.
Now calculate the ratio of to
The common ratio was found to be Since the lower limit of the sigma notation is and the upper limit of the sigma notation is there are terms in the series in total, so The sum can be calculated with these values. To do so, substitute all the values into the formula for and simplify.
Evaluate right-hand side
Pop Quiz

Calculating Finite Geometric Series

Calculate the sum of all the terms of the given finite geometric series written in summation notation. Remember that the formula for the sum of a geometric series can be used rather than adding the terms one by one.

Calculating finite geometric series in summation notation
Discussion

Infinite Geometric Series

Let and be the first term and the common ratio, respectively, of a geometric sequence with terms, where For an infinite series, if the common ratio is greater than and less than — in other words, if — then the sum can be found by using the following formula.

This means that the sum converges on a number. If the common ratio is less than or equal to or greater than or equal to — if — then the sum diverges. In such cases, there is no sum for the infinite geometric series.

Proof

Infinite Sum
To derive the formula for the sum of an infinite geometric series with the standard formula for the finite geometric series will be considered.
Since is a number between and the value of becomes very small as the value of increases. In other words, it gets closer to as approaches infinity.
Therefore, can be substituted into the standard formula and the resulting equation simplified.
The formula for the sum of an infinite geometric series with has been derived.

Example

Solving Examples of Geometric Series

In the next math lesson, Tearrik's math teacher continued with the topic geometric series and this time she introduced the first few terms of two different geometric series as examples.
Two example of series shown in the computer screen. Series 1: 1+2/3+ 4/9+ 8/27 + 16/81 + ··· Series 2: -10/4+50/16-250/64+1250/256-···
a Determine whether the geometric series in Example I converges or diverges. If it converges, find its sum.
b Determine whether the geometric series in Example II converges or diverges. If it converges, find its sum.

Answer

a Does the series converge or diverge? Converge

Sum:

b Does the series converge or diverge? Diverge

Sum: No sum.

Hint

a Consider the common ratio of the series. When does a series converge or diverge?
b What is the common ratio of the given series?

Solution

a To determine whether the sum of the series converges to a number or if the series does not have a sum, start by looking at the common ratio between the terms of the given series in Example I.
Terms of the first example series
As shown, the common ratio is Since the absolute value of less than it means that the sum of the series does converge to a number.
Therefore, it is possible to find its sum by using the formula for the sum of an infinite series.
To find the sum, substitute and into the formula and evaluate.
Evaluate right-hand side
The sum of the infinite geometric series given in Example I is
b Once again, start by identifying the common ratio of the given series. One way to find the common ratio is to divide the second term by the first term.
Simplify
The common ratio of the given series in Example II is
Terms of the second example series
Now the absolute value of the common ratio will be found.
Since the absolute value of the common ratio is greater than the series diverges. Therefore, it is not possible to find a sum for this series.
Pop Quiz

Does It Converge or Diverge?

Determine whether the given infinite geometric series converge or diverge. Remember that if the common ratio then the infinite series converges to a number, and that if then the series diverges.

Applet showing some series
Discussion

Partial Sum of an Infinite Geometric Series

If the common ratio of an infinite geometric series is less than or equal to or greater than or equal to the sum of the series does not exist. However, it is possible to find a partial sum or the sum of the first several terms in the series. This partial series can be thought of as a finite series. As such, its sum can be found using the formula for a finite geometric series.

Example

Calculating the Distance Traveled by the Ball

After recovering from his illness, Tearrik returns to school and continues to play basketball with his best friend Tadeo. Suppose that after the ball hits the rim of the basket, the ball falls meters and rebounds to of the height of the previous bounce.

the path of the ball after hitting the rim
a Find the total vertical distance traveled by the ball before it comes to rest.
b What would the sum of the vertical distance traveled by the ball be if Tadeo caught the ball at the top of the fourth bounce? Round the answer to two decimal places.

Hint

a The vertical distance traveled by the ball for one bounce will be times of the previous bounce since the ball rises and then falls the same distance.
b Use the formula for the partial sum of an infinite series.

Solution

a It is given that after each bounce, the ball rises of the height of the previous bounce. Since the initial height of the ball is meters, the first bounce will be times the initial height.
For the second bounce, the height of the ball will be times the first bounce, meters.
The heights of the other bounces can be written by considering this pattern.
the path of the ball after hitting the rim
Because the ball rises and then falls the same distance after each bounce, the vertical distance traveled is times of the previous bounce. Since the initial height of the ball is meters, the total distance traveled by the ball can be written as follows.
Since a common ratio exists between the distances that the ball travels, the distances traveled starting with the first bounce represent a geometric series. This sum is considered infinite because it is assumed that the ball could continue to bounce in increasingly small increments forever. Now, express the sum using summation notation.
Note that even though the sum of the series is infinite, it can still be found by using the formula for the sum of an infinite series because the absolute value of the common ratio is less than
The first term of the infinite geometric series is and the common ratio is Now, substitute these values into the formula!
Then, evaluate the right hand side.
Evaluate right-hand side
The sum of the series is Now the initial meter height of the ball will be added.
This means that the ball will have traveled meters when it comes to rest.
b This time the ball is caught at the top of the fourth bounce. To find the vertical distance traveled by the ball, start by considering the vertical distance traveled for each bounce until the end of the third bounce. Then, add the fourth vertical distance only once since the ball will rise and but be caught before it falls again.
Notice that this sum represents a partial sum of the infinite series considered in Part A. To calculate this partial sum, the formula for the sum of a finite geometric series will be used, as the sum is calculated up until
With this in mind, substitute and into the formula and evaluate the result.
Evaluate right-hand side
Now, add the fourth vertical distance to this sum.
Finally, the initial meter height will be added.
The ball will travel vertically about meters in total until Tadeo catches it at the top of the fourth bounce.
Example

Who Will Save More Money?

Now that they are completely recovered, Tearrik and Tadeo decide to save money so they can go to the NBA finals next year, months from now. They start chatting about their own ways to save money.

Conversation between Tadeo and Tearrik about the way of saving money
Who will save more money in months?

Hint

Recall that a series is geometric if it has a common ratio.

Solution

To determine who will save more money in months, the total amounts saved will be calculated one at a time.

Tadeo's Savings

Tadeo is planning to save dollars every month. This means that his savings will increase linearly by a constant rate of dollars. Therefore, multiply dollars by months to find the total amount of money he will have saved in months.
After months, Tadeo will have saved dollars.

Tearrik's Savings

Tearrik will start by saving only the first month. Then, he will increase the amount of money he saves every month by doubling the previous amount.

increase in the amount of money saved
Since there is a common ratio between the amount of money saved each month, this sum represents a geometric series. Therefore, the formula for the sum of a finite geometric series will be used to calculate the total amount of money Tearrik will have saved in months.
Now, substitute and into this formula and simplify.
Evaluate right-hand side