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# Geometric Sequences & Series 8.3 PowerPoint PPT Presentation

Geometric Sequences & Series 8.3. JMerrill, 2007 Revised 2008. Sequences. A Sequence: Usually defined to be a function Domain is the set of positive integers Arithmetic sequence graphs are linear (usually) Geometric sequence graphs are exponential. Geometric Sequences.

Geometric Sequences & Series 8.3

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## Geometric Sequences & Series8.3

JMerrill, 2007

Revised 2008

### Sequences

• A Sequence:

• Usually defined to be a function

• Domain is the set of positive integers

• Arithmetic sequence graphs are linear (usually)

• Geometric sequence graphs are exponential

### Geometric Sequences

• GEOMETRIC - the ratio of any two consecutive terms in constant.

• Always take a number and divide by the preceding number to get the ratio

• 1,3,9,27,81……….

ratio = 3

• 64,-32,16,-8,4……

ratio = -1/2

• a,ar,ar2,ar3………

ratio = r

2

-2/3

• Arithmetic

• Geometric

• Neither

• Arithmetic

• Geometric

• Neither

• Arithmetic

• Geometric

• Neither

### Example

• Write the first six terms of the geometric sequence with first term 6 and common ratio 1/3.

### Formulas for the nth term of a Sequence

• Geometric:an=a1 * r (n-1)

• To get the nth term, start with the 1st term and multiply by the ratio raised to the (n-1) power

n = THE TERM NUMBER

### Example

• Find a formula for an and sketch the graph for the sequence 8, 4, 2, 1...

• Arithmetic or Geometric?

• r = ?

• an = a1 (r (n-1) )

• an = 8 * ½ (n-1)

n = THE TERM NUMBER

### Using the Formula

• Find the 8th term of the geometric sequence whose first term is -4 and whose common ratio is -2

• an=a1 * r (n-1)

• a8=-4 * (-2) (8-1)

• a8 = -4(-128) = 512

### Example

• Find the given term of the geometric sequence if a3 = 12, a6 =96, find a11

• r = ? Since a1 is unknown. Use given info

• an = a1 * r (n-1)an = a1 * r (n-1)

• a3 = a1 * r2a6 = a1 * r5

• 12 = a1 *r296 = a1 *r5

### Sum of a Finite Geometric Series

• The sum of the first n terms of a geometric series is

Notice – no last term needed!!!!

### Example

• Find the sum of the 1st 10 terms of the geometric sequence: 2 ,-6, 18, -54

What is n? What is a1? What is r?

That’s It!

### Infinite Geometric Series

• Consider the infinite geometric sequence

• What happens to each term in the series?

• They get smaller and smaller, but how small does a term actually get?

Each term approaches 0

### Partial Sums

• Look at the sequence of partial sums:

1

0

What is happening to the sum?

It is approaching 1

### Here’s the Rule

So, if -1 < r < 1, then the series will converge. Look at the series given by

Since r = , we know that the sum

is

The graph confirms:

### Converging – Has a Sum

If r > 1, the series will diverge. Look at 1 + 2 + 4 + 8 + ….

Since r = 2, we know that the series grows without bound and has no sum.

The graph confirms:

### Example

• Find the sum of the infinite geometric series 9 – 6 + 4 - …

• We know: a1 = 9 and r = ?

### You Try

• Find the sum of the infinite geometric series 24 – 12 + 6 – 3 + …

• Since r = -½

### Example

• Ex: The infinite, repeating decimal 0.454545… can be written as the infinite series

0.45 + 0.0045 + 0.000045 + …

• What is the sum of the series? (Express the decimal as a fraction in lowest terms)

### You Try

• Express the repeating decimal, 0.777…, as a rational number (hint: the sum!)

### You Try, Part Deux

• Find the first three terms of an infinite geometric sequence with sum 16 and common ratio

### Last Example

• Find the following sum:

• What’s the first term?

• What’s the second term?

• Arithmetic or Geometric?

• What’s the common ratio?

• Plug into the formula…

12

24

2

### Can You Do It???

• Find the sum, if possible, of

• 8