8.3 Analyzing Geometric Sequences and Series

Identifying Geometric Sequences In a geometric sequence, the ratio of any term to the previous term is constant. The constant ratio is called the common ratio and is denoted by r. For example, in the geometric sequence 1, 2, 4, 8,… the common ratio is 2.

Example 1: Tell if the sequence below is geometric. 6, 12, 20, 30,… Find the ratios of consecutive terms. The ratios are not constant, so the sequence is not geometric.

Example 2: Tell if the sequence below is geometric. 256, 64, 16, 4,… Find the ratios of consecutive terms. The ratios are constant, so the sequence is geometric.

Geometric Sequence Example of geometric sequence: A sequence of numbers is called a geometric sequence if the ratio of any two consecutive terms is constant. This ratio is called a common ratio r. Example of geometric sequence:

Example 3: Write a rule for the nth term of the sequence below. 5, 15, 45, 135,… The sequence is geometric with first term = 5 and the common ratio So, a rule for the nth term is

Example 4: In a geometric sequence, and the common ratio is r = 2 Example 4: In a geometric sequence, and the common ratio is r = 2. Write a rule for the nth term. Then graph the first six terms of the sequence. Given: Geometric Sequence STEP 1: Use the general rule to find the first term. STEP 2: Write a rule for the nth term. STEP 3: Create a table of values and then plot the points. n 1 2 3 4 5 6 1.5 12 24 48

Example 5: GIVEN: Geometric Sequence

Finding Sums of Finite Geometric Series The expression formed by adding the terms of a geometric sequence is called a geometric series. The sum of the first n terms of a geometric series is denoted by . The sum of the first n terms of a geometric series with common ratio is

Proof of Sum of Geometric Series Formula

Example 6: Find the sum STEP 1: Find the first term and the common ratio. Identify first term. Identify common ratio. STEP 2: Find the sum. GIVEN: Geometric Series Write rule for . Substitute 4 for and 3 for r.

Example 7: Find the sum . STEP 1: Find the first term and the common ratio. Identify first term. Identify common ratio. STEP 2: Find the sum. GIVEN: Geometric Series Substitute values and solve.

Example 8: You can calculate the monthly payment M (in dollars) for a loan using the formula where L is the loan amount (in dollars), i is the monthly interest rate (in decimal form), and t is the term (in months). Calculate the monthly payment on a 5-year loan for $20,000 with an annual interest rate of 6%. GIVEN: L = $20000 t = time in months i = monthly interest rate (decimal form) Annual interest rate = 0.06 M(5 years) = ?

GIVEN: GIVEN: L = $20000 t = time in months i = monthly interest rate (decimal form) Annual interest rate = 0.06 M(5 years) = ? STEP 1: Substitute for L, i, and t. The loan amount is L = 20,000, the monthly interest rate is i = 0.06/12 = 0.005, and the term is t = 5(12)=60. STEP 2: Notice that the denominator is a geometric series with first term 1/1.005 and common ratio 1/1.005.