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Introduction to Geometric Sequences and Series
23 May 2011
Investigation:
Find the next 3 terms of each sequence:{3, 6, 12, 24, …}{32, 16, 8, 4, …}
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
Finding the Common Ratio
Find r, the common ratio:
1. {3, 9, 27, 81, …}
2.
2nu
ur
1n
n
• Divide any term by its previous term
,...
16
5,
8
5,
4
5,
2
5
Your Turn: Find r, the common ratio:
1. {0.0625, 0.25, 1, 4, …}
2. {-252, 126, -63, 31.5, …}
3.
,...
192
2,
48
2,
12
2,
3
2
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
1n
n
u
ur
Your Turn: Classifying Sequences Determine if each sequence is arithmetic,
geometric, or neither:
1. {2, 7, 12, 17, 22, …}
2. {-6, -3.7, -1.4, 9, …}
3. {-1, -0.5, 0, 0.5, …}
4. {2, 6, 18, 54, 162, …}
5.
,...16
3,
8
3,
4
3,
2
3,3
Recursive Form of a Geometric Sequence
un = run–1 n ≥ 2
nth term n–1th termcommon ratio
Example #1 u1 = 2, u2 = 8
1. Write the recursive formula
2. Find the next two terms
Example #2 u1 = 14, u2 = 39
1. Write the recursive formula
2. Find the next two terms
Your Turn: For the following problems, write the
recursive formula and find the next two terms:
1. u1 = 4, u2 = 4.25
2. u1 = 90, u2 = -94.5
3. ,
2
1u1
16
1u2
Explicit Form of a Geometric Sequence
un = u1rn–1 n ≥ 1
nth term1st term
common ratio
Example #1
u1 = 2,
1. Write the explicit formula
2. Find the next three terms
3. Find u12
5
1r
Example #2 u1 = 6, u2 = 18
1. Write the explicit formula
2. Find the next three terms
3. Find u12
Your Turn: For the following problems, write the
explicit formula, find the next three terms, and find u12
1. u1 = 5, r = -¼
2. u1 = 5, u2 = -20
3. u1 = 144, u2 = 72
Partial Sum of a Geometric Sequence
1rr1
r1uu
k
1
k
1nn
Example #1 k = 9, u1 = -1.5, r = -½
Example #2 k = 6, u1 = 1, u2 = 5
Example #3 k = 8, ,
4
3u1
4
1u2
Your Turn: Find the partial sum:
1. k = 6, u1 = 5, r = ½
2. k = 8, u1 = 9, r = ⅓
3. k = 7, u1 = 3, u2 = 6
4. k = 8, u1 = 24, u2 = 6