Introduction Terms of geometric sequences can be added together if needed, such as when calculating the total amount of money you will pay over the life of a loan that charges interest. When a group of some number of terms from a geometric sequence are added together, that group is called a geometric series. The number of terms in that group could go on forever, or the terms could be finite— that is, limited to a certain number. A finite geometric series is the sum of a specified number of terms of a geometric sequence. In this lesson, we will learn different ways to write and evaluate a finite geometric series : Sum of a Finite Geometric Series

Key Concepts A geometric series is the sum of some number of terms from a geometric sequence. Geometric sequences and series are related: Both have a common ratio. A sequence is a listing of the terms in a geometric progression. A series is the sum of part of that list : Sum of a Finite Geometric Series

Key Concepts, continued A finite series finds the sum of a specified number of the terms from the sequence. Compare the first five terms of the geometric sequence {1, 2, 4, 8, 16, …} with the same terms written as a geometric series: Sequence: 1, 2, 4, 8, 16 Series: : Sum of a Finite Geometric Series

Key Concepts, continued Summation notation is a symbolic way to represent the sum of a sequence. A common way to write summation notation for a geometric series is, where n is a positive integer that represents the number of terms in this series, a 1 is the first term, r is the common ratio, and k is the number of the term : Sum of a Finite Geometric Series

Key Concepts, continued This notation is read as “take the sum from 1 to n of the series a 1 times r to the k – 1.” The symbol is the uppercase Greek letter sigma. In this context, it means “to take the sum.” The n represents the number of iterations to complete in the sum, or the final term in the sum. For example, if n = 3, you will add 3 of the terms in the series. k = 1 shows the starting value of the iterations, or the number of the term to start with in the sum : Sum of a Finite Geometric Series

Key Concepts, continued The diagram below illustrates the parts of this summation notation formula : Sum of a Finite Geometric Series

Key Concepts, continued Finite series have a definite ending term—they do not go on forever. Infinite series go to infinity. These will be explored later. Summing terms can be tedious. To make this process easier, mathematicians developed a formula for finding the sum of terms in a series : Sum of a Finite Geometric Series

Key Concepts, continued The sum formula for a finite geometric series from 1 to n is, where S n is the sum, a 1 is the first term, r is the common ratio, and n is the number of terms. Notice that the right side of the formula is a fraction. The denominator of a fraction cannot equal 0. Therefore, in this formula, r ≠ : Sum of a Finite Geometric Series

Key Concepts, continued Loan Payments and Sums of Finite Geometric Series The sum formula of a finite geometric series affects everyone who makes payments on a loan, whether they know it or not. When you take out a loan for a home or car, and pay interest on that loan, the total amount you will have paid by the time you make your last payment is actually the sum of a finite geometric series : Sum of a Finite Geometric Series

Key Concepts, continued Let’s say you take out a loan for a home. You will take out a loan for the price of the home, and make monthly payments to the lender. The bank that loaned you the money will charge interest each month until the loan is paid off. At the beginning of your loan, most of your payment goes to interest, because your principal is large. As the years pass, more of your payment goes to the principal and less goes to interest. This kind of loan is called an amortized loan : Sum of a Finite Geometric Series

Key Concepts, continued The formula for the sum of an amortized loan is a geometric series. The formula is used to calculate how much money will have been paid in principal and interest by the time a loan is paid off. The formula is given by where P is the principal (that is, the loan amount), A is the monthly payment amount, and i is the monthly interest based on the annual percentage rate (APR) divided by : Sum of a Finite Geometric Series

Key Concepts, continued The table on the next slide shows how this amortization formula relates to the general summation notation shown earlier : Sum of a Finite Geometric Series

: Sum of a Finite Geometric Series Key Concepts, continued Comparing Summation Notation to the Amortization Formula Complete formula Sum of the series First term Common ratio Placement of the term in the sequence ark – 1 A

Common Errors/Misconceptions misinterpreting the parts of the summation notation forgetting that anything raised to the 0 power is equal to 1 skipping iterations substituting numbers into the summation formula incorrectly : Sum of a Finite Geometric Series

Guided Practice Example 1 Expand the series shown in the given summation notation : Sum of a Finite Geometric Series

Guided Practice: Example 1, continued 1.Determine the number of terms in the expanded series. The number shown above the sigma symbol represents the final term of the sum (n). Here, n = 6. Therefore, the expanded series will have 6 terms : Sum of a Finite Geometric Series

Guided Practice: Example 1, continued 2.Substitute values for k. The portion of the summation notation that shows the value of each term in the sequence is. Since n = 6, write the first 6 terms of the series, each time substituting one of the counting numbers 1 through 6 for k in the exponent : Sum of a Finite Geometric Series

Guided Practice: Example 1, continued Set the expansion up as an equation equal to the original notation, adding the rewritten terms together to the right of the equal sign: : Sum of a Finite Geometric Series

Guided Practice: Example 1, continued 3.Simplify and apply the exponents. Start by simplifying each exponent. Then, apply each exponent to the fraction in parentheses, as shown: : Sum of a Finite Geometric Series

Guided Practice: Example 1, continued 4.Simplify the terms. The expanded series is : Sum of a Finite Geometric Series ✔

Guided Practice: Example 1, continued : Sum of a Finite Geometric Series

Guided Practice Example 2 Sum the series by expansion : Sum of a Finite Geometric Series

Guided Practice: Example 2, continued 1.Expand the series. k = 1 and n = 5; therefore, the series starts with the first term of the sequence and ends with the fifth term. Write the first 5 terms of the series, each time substituting one of the counting numbers 1 through 5 for k in the exponent as shown: : Sum of a Finite Geometric Series

Guided Practice: Example 2, continued Simplify the exponents for each term. Apply each exponent to its term. Notice that each term has a factor of : Sum of a Finite Geometric Series

Guided Practice: Example 2, continued 2.Factor the expression and simplify : Sum of a Finite Geometric Series Series with exponents applied Factor out 2 from the terms in the series. Sum the terms. Multiply.

Guided Practice: Example 2, continued The sum of the series containing the first 5 terms of the sequence is : Sum of a Finite Geometric Series ✔

Guided Practice: Example 2, continued : Sum of a Finite Geometric Series