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Geometric Sequences & Series 8.3PowerPoint Presentation

Geometric Sequences & Series 8.3

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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

Is the Sequence 3, 8, 13, 18…

- Arithmetic
- Geometric
- Neither

Is the Sequence 2, 5, 10, 17…

- Arithmetic
- Geometric
- Neither

Is the Sequence 8, 12, 18, 27…

- 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 *r2 96 = 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

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 SumIf 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:

Diverging – Has NO SumExample 4 + 8 + ….

- Find the sum of the infinite geometric series 9 – 6 + 4 - …
- We know: a1 = 9 and r = ?

You Try 4 + 8 + ….

- Find the sum of the infinite geometric series 24 – 12 + 6 – 3 + …
- Since r = -½

Example 4 + 8 + ….

- 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 4 + 8 + ….

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

You Try, Part Deux 4 + 8 + ….

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

Last Example 4 + 8 + ….

- 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??? 4 + 8 + ….

- Find the sum, if possible, of
- 8

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