1. Introduction to Geometric Sequences and Series
23 May 2011

2. Investigation: Find the next 3 terms of each sequence:
{3, 6, 12, 24, …}
{32, 16, 8, 4, …}

3. Geometric Sequences
Sequences that increase or decrease by multiplying the previous term by a fixed number
This fixed number is called r or the common ratio

4. Finding the Common Ratio
Divide any term by its previous term
Find r, the common ratio:
{3, 9, 27, 81, …}

5. Your Turn: Find r, the common ratio: {0.0625, 0.25, 1, 4, …}
{-252, 126, -63, 31.5, …}

6. Arithmetic vs. Geometric Sequences
Arithmetic Sequences
Increases by the common difference d
Addition or Subtraction
d = un – un–1
Geometric Sequences
Increases by the common ratio r
Multiplication or Division

7. Your Turn: Classifying Sequences
Determine if each sequence is arithmetic, geometric, or neither:
{2, 7, 12, 17, 22, …}
{-6, -3.7, -1.4, 9, …}
{-1, -0.5, 0, 0.5, …}
{2, 6, 18, 54, 162, …}

8. Recursive Form of a Geometric Sequence
un = run–1 n ≥ 2
nth term
n–1th term
common ratio

9. Example #1 u1 = 2, u2 = 8 Write the recursive formula
Find the next two terms

10. Example #2 u1 = 14, u2 = 39 Write the recursive formula
Find the next two terms

11. Your Turn:
For the following problems, write the recursive formula and find the next two terms:
u1 = 4, u2 = 4.25
u1 = 90, u2 = -94.5

12. Explicit Form of a Geometric Sequence
common ratio
un = u1rn–1 n ≥ 1
nth term
1st term

13. Example #1 u1 = 2, Write the explicit formula
Find the next three terms
Find u12

14. Example #2 u1 = 6, u2 = 18 Write the explicit formula
Find the next three terms
Find u12

15. Your Turn:
For the following problems, write the explicit formula, find the next three terms, and find u12
u1 = 5, r = -¼
u1 = 5, u2 = -20
u1 = 144, u2 = 72

16. Partial Sum of a Geometric Sequence

17. Partial Sum of a Geometric Sequence
upper bound (ending term)
common ratio
lower bound (starting term)

18. Example #1
k = 9, u1 = -1.5, r = -½

19. Example #2
k = 6, u1 = 1, u2 = 5

20. Example #3
k = 8,

21. Your Turn: Find the partial sum: k = 6, u1 = 5, r = ½
k = 7, u1 = 3, u2 = 6
k = 8, u1 = 24, u2 = 6