Section 10 3 Geometric Sequences Series Objectives Find

Section 10. 3: Geometric Sequences & Series Objectives: -Find nth terms and geometric means of geometric sequences - Find sum of n terms of geometric series and the sums of infinite geometric series

Definitions Geometric Sequence: when the ratio between consecutive terms is a constant. Common Ratio: denoted r, the constant ratio between terms. *To find r, divide any term by its previous term. *To find the next term in the sequence, MULTIPLY the given term by the common ratio.

Example 1 Determine the common ratio and find the next three terms of the geometric sequence: A) – 6, 9, – 13. 5, …. B) 243 n – 729, – 81 n + 243, 27 n – 81, ….

Example 1 �

Explicit Formula �

Example 2: Write an explicit formula and a recursive formula for finding the nth term in the sequence: A) – 1, 2, – 4, …. B) 3, 16. 5, 90. 75, …

Example 3: A) Find the 11 th term of the geometric sequence – 122, 115. 9, – 110. 105, …. B) Find the 25 th term of the geometric sequence 324, 291. 6, 262. 44, ….

Example 3: �

Exit Slip � Find the 14 th term of the geometric sequence 3, – 9, 27, ….

Example 4: A) REAL ESTATE A couple purchased a home for $225, 000. At the end of each year, the value of the home appreciates 3%. Write an explicit formula for finding the value of the home after n years. What is the value of the home after the tenth year?

Example 4: B) BOAT Jeremy purchased a boat for $12, 500. By the end of each year, the value of the boat depreciates 4%. Write an explicit formula for finding the value of the boat after n years. Then find the value of the boat in 20 years.

The graphs of terms of a geometric sequence lie on a curve, as shown. A geometric sequence can be modeled by an exponential function in which the domain is restricted to natural numbers.

Geometric Mean � If two nonconsecutive terms of a geometric sequence are known, the terms can be calculated. � General steps to do so: �Find the common ratio using the nth term formula for geometric sequences. �Use r to find the geometric means (next terms)

Example 5: A) Write a sequence that has three geometric means between 264 and 1. 03125.

Example 5: B) Write a geometric sequence that has two geometric means between 20 and 8. 4375.

Definitions Geometric Series: the sum of the terms of a geometric sequence.

Example 6: A) Find the sum of the first eleven terms of the geometric series 4, – 6, 9, …. B) Find the sum of the first n terms of a geometric series with a 1 = – 4, an = – 65, 536, and r = 2.

Example 6: C) Find the sum of the first 8 terms of the geometric series 8 + 36 + 162 + ….

Example 7: A) Find B) Find

Infinite Geometric Series �

Example 8: �

Example 9: Express the series using sigma notation. Then find the indicated sum. a) 3 + 12 + 48 + … + 3072 b) 0. 2 – 1 + 5 - … - 625

Example 9: �

Example 10: a) In a geometric sequence a 2 = – 8 and a 7 = 8192. Find S 10.

Example 10: b) In a geometric sequence a 3 = 48 and a 8 = 1536. Find S 8.

Exit Slip: �