1. Sec 11.3 Geometric Sequences and Series Objectives: To define geometric sequences and series. To define infinite series. To understand the formulas for sums of finite and infinite geometric series.

2. An arithmetic sequence is defined when we repeatedly add a number, d, to an initial term. A geometric sequence is generated when we start with a number, a 1, and repeatedly multiply that number by a fixed nonzero constant, r, called the common ratio.

3. Definition of a Geometric Sequence A geometric sequence is a sequence of the form a, ar, ar 2, ar 3, ar 4,.. The number a is the first term, and r is the common ratio of the sequence. The n th term of a geometric sequence is given by

4. Ex. 1 Find the first four terms and the n th term of the geometric sequence with a = 2 and r = 3. Since a = 2 and r = 3, we can plug into the nth term formula to get the terms.

5. Ex 2. Find the eighth term of the geometric sequence 5, 15, 45,…

6. Ex 3. The third term of a geometric sequence is 63/4, and the sixth term is 1701/32. Find the fifth term. Using the nth term formula and the two given terms we get a system of equations. If we divide the equations we get the following: And so, we get: Since we have the sixth term, just divide it by r and you will get the fifth term.

7. Partial Sums For the geometric sequence a, ar, ar 2, ar 3, ar 4,..., ar n–1,..., the nth partial sum is:

8. Ex 4. Find the sum of the first five terms of the geometric sequence 1, 0.7, 0.49, 0.343,...

9. Ex. 5 Find the sum. Plug 1 in for k and you will get a = -14/3. Then plug in a and r into the sum formula.

10. You do not want to use decimals. All answers will be exact (which means they will be given as fractions.) We do not want to get rounded answers. Use the fraction key on your calculator to help you get the numbers you see here.

11. HW #3 Finite Geometric Series Wkst odds (even extra credit)

12. Infinite Series An expression of the form a 1 + a 2 + a 3 + a 4 +... is called an infinite series.

13. Let’s take a look at the partial sums of this series.

14. Sum of an Infinite Geometric Series If | r | < 1, the infinite geometric series a + ar + ar 2 + ar 3 + ar 4 +... + ar n–1 +... has the sum

15. Ex 6 Find the sum of the infinite geometric series

16. HW #4 Infinite Series Wkst odds (evens extra credit)