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