Objectives: 1. Recognize a geometric sequence 2. Find a common ratio 3. Graph a geometric sequence 4. Write a geometric sequence recursively and explicitly 5. Find the nth term of a geometric sequence 6. Find partial sums of a geometric sequence

 A geometric sequence, which is sometimes called a geometric progression, is a sequence in which terms are found by multiplying a preceding term by a nonzero constant.  In a geometric sequence, the quotient of consecutive terms is the constant r called the common ratio. For example, in the geometric sequence {80, 40, 20, 10, …}, the common ratio r = ½ because each consecutive term is multiplied by ½.

In a geometric sequence {u n }, u n = r ∙ u n−1 For some u 1 and some nonzero constant r and all n ≥ 2

Example #1 Recognizing a Geometric Sequence  Are the following sequences geometric? If so, what is the common ratio? Write each sequence as a recursive function. A. B.

Example #2 Graph of a Geometric Sequence  Find the common ratio of the geometric sequence with u 1 = 3 and u 2 = 6. List the first five terms of the sequence, write the sequence as a recursive function, and graph the function. Term #NowNext 132(3) = 6 262(6) = (12) = (24) =

Example #3 Writing a Geometric Sequence in Explicit Form  Confirm that the sequence defined by u n = 3u n−1 with u 1 = 6 can also be expressed as u n = 6(3) n−1 by listing the first five terms produced by each form. Term #NowNext 163(6) = (18) = (54) = Term #Now 16 26(3) 2 − 1 = 18 36(3) 3 − 1 = 54 46(3) 4 − 1 = 162 u n = 3u n−1 u n = 6(3) n−1

If {u n } is a geometric sequence with common ratio r, then for all n ≥ 1, u n = u 1 r n−1

Example #4 Explicit Form of a Geometric Sequence  Write the explicit form of a geometric sequence where the first two terms are 3 and − ¾, and find the first five terms of the sequence. Term #Now (− 1/4) 2−1 = −3/4 33( −1/4) 3−1 = 3/16 4 3(−1/4) 4−1 = −3/64 53( −1/4) 5−1 = 3/256

Example #5 Finding a term of a geometric sequence.  What is the 100 th term of a geometric sequence whose first three terms are 3, 9, and 27.

Example #6 Explicit Form of a Geometric Sequence  The third and sixth terms of a geometric sequence are 80 and −5120. Find the explicit form of the sequence. Just like with the arithmetic sequences, the explicit formula for geometric sequences can be generalized where u n is the higher term number and u p is the lower term number.

Example #6 Explicit Form of a Geometric Sequence  The third and sixth terms of a geometric sequence are 80 and −5120. Find the explicit form of the sequence.  What is the 20 th term?

Partial Sums of a Geometric Sequence The k th partial sum of the geometric sequence {u n } with common ratio r ≠ 1 is Just like with arithmetic sequences, a formula can be used to quickly calculate summations of geometric sequences without finding every term. u 1 is the first term, r is the common ratio, and k is the number of terms.

Example #7 Partial Sum  Find the sum. To use this formula you must first find the common ratio r.

Example #8 Application of Partial Sum  A ball is dropped from a height of 10 feet. It hits the floor and bounces to a height of 7.5 feet. It continues to bounce up and down. On each bounce it rises to ¾ of the height of the previous bounce. How far has it traveled (both up and down) when it hits the floor for the ninth time? This pattern can be modeled by a geometric sequence: This time since the initial height of 10 feet isn’t a bounce we will exclude it from our sequence, but add it to the total distance in the end.