Geometric Sequences and Series OBJECTIVES: Determine if a Sequence is Geometric Find a Formula for a Geometric Sequence Find the Sum of a Geometric Sequence Find the Sum of a Geometric Series

A sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence. Common Ratio, r, is found by dividing any term after the first term by the term that directly precedes it. General Term of a Geometric Sequence: the nth term (the general term) of a geometric sequence with first term and common ratio Explicit Formula: Recursive Formula: Geometric Sequence

EX 1: Find the common ratio and write out the first four terms of the geometric sequence

EX 2: Find the nth term of the geometric sequence whose initial term and common ratio are given. What is the fifth term? Geometric Sequence

EX 3: Find the eighth term of the geometric sequence whose first term is -4 and whose common ratio is -2 Geometric Sequence

Arithmetic, Geometric or Neither EX 4: Determine the type of sequence A) B) C) Arithmetic, Geometric or Neither

The Sum of the First n terms of a Geometric Sequence The sum of the first terms of a geometric sequence is given by: In which is the first term and is the common ratio The Sum of the First n terms of a Geometric Sequence

EX 5: Find the sum of the first 18 terms of the geometric sequence: Sum of n terms

A) B) EX 6: Find the Sum

The Sum of an Infinite Geometric Series If , then the sum of the infinite geometric series in which is the first term and is the common ratio is given by: If , then the infinite series does not have a sum The Sum of an Infinite Geometric Series

EX 7: Find the sum of each infinite geometric series B) EX 7: Find the sum of each infinite geometric series