1. Geometric Sequences and Series
Chapter 8.3
Geometric Sequences and Series

2. What You’ll Learn
Recognize, write, and find the nth terms of geometric sequences
Find nth partial sums of geometric sequences
Find sums of infinite geometric series
Use geometric sequences to model and solve real-life problems

3. Geometric Sequences
Definition of Geometric Sequence A sequence is geometric if the ratios between consecutive terms are the same. So, the sequence 𝑎 1 , 𝑎 2 , 𝑎 3 , 𝑎 4 , …, 𝑎 𝑛 , … is geometric if there is a number 𝑟 such that 𝑎 2 𝑎 1 = 𝑎 3 𝑎 2 = 𝑎 4 𝑎 3 =…=𝑟 The number 𝑟 is the common ratio of the geometric sequence.

4. Geometric Sequences
Example 1. Determine whether or not the sequence is geometric . If it is find the common ratio. a) 5, 8, 11, 14, … b) 1, 1 2 , 1 4 , 1 8 , … c) 1 5 , 2 7 , 3 9 , 4 11 , …

5. Geometric Sequences
Example 2. Write the first five terms of the geometric sequence. a) 𝑎 1 =6, 𝑟=3 b) 𝑎 1 =1, r= 1 2 c) 𝑎 1 =1, 𝑟=𝑒

6. Geometric Sequences
The 𝒏th Term of an Geometric Sequence The 𝑛th term on an geometric sequence can be developed as follows 𝑎 1 = 𝑎 1 𝑎 2 = 𝑎 1 𝑟 𝑎 3 = 𝑎 2 𝑟= 𝑎 1 𝑟 𝑟= 𝑎 1 𝑟 2 𝑎 4 = 𝑎 3 𝑟= 𝑎 1 𝑟 2 𝑟= 𝑎 1 𝑟 3 ⋮ 𝑎 𝑛 = 𝑎 1 𝑟 𝑛−1

7. Geometric Sequences
Example 3. Find the 𝑛th term of the arithmetic sequence with common ratio 2 and first term 5.

8. Geometric Sequences
Example 4. Find the 𝑛th term of the geometric sequence.
a) 𝑎 1 =4, 𝑟= 1 2 , 𝑛=10
b) 𝑎 1 =86 𝑟=− 1 3 , 𝑛=12
c) 𝑎 1 =500, 𝑟=1.02, 𝑛=14

9. Geometric Sequences
Example 5. Find the 20th term of the geometric sequence 1, 3, 9, 27, …

10. Geometric Sequences
Example 6. Write the first five terms of the geometric sequence. Find the common ratio and write the 𝑛th term of the sequence as a function of 𝑛. a) 𝑎 1 =19, 𝑎 𝑘+1 = 1 2 𝑎 𝑘 b) 𝑎 1 =6, 𝑎 𝑘+1 = − 3 2 𝑎 𝑘

11. Geometric Sequences
The Sum of a Finite Geometric Sequence (also called an Geometric Series) The sum of a finite arithmetic with 𝑛 terms is given by 𝑆 𝑛 = 𝑎 1 − 𝑎 1 𝑟 𝑛 1−𝑟 or 𝑆 𝑛 = 𝑎 1 1− 𝑟 𝑛 1−𝑟

12. Arithmetic Sequences
Example 8. Find the sum. 𝑛=1 9 2 𝑛−1

13. Arithmetic Sequences
Example 9. Find the sum. 𝑛= 𝑛

14. Geometric Series
The Sum of a Infinite Geometric Series If 𝑟 <1, then the infinite geometric series 𝑎 1 + 𝑎 1 𝑟+ 𝑎 1 𝑟 2 + 𝑎 1 𝑟 3 +…+ 𝑎 1 𝑟 𝑛−1 +… has the sum 𝑆= 𝑖=0 ∞ 𝑎 1 𝑟 𝑖 = 𝑎 1 1−𝑟 If 𝑟 ≥1, the series does not have a sum

15. Geometric Series
Example 10. Find the sum of the infinite geometric series, if possible. 𝑛=0 ∞ 𝑛

16. Geometric Series
Example 11. Find the sum of the infinite geometric series, if possible. 𝑛=1 ∞ 𝑛

17. Applications
Example 12. A principal of $2500 is invested at 4% interest. Find the amount after 20 years if the interest is compounded (Hint: You’ll need the formula 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 ) a) annually b) quarterly