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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.

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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


What is the ratio of 4, 8, 16, 32…

2


What is the ratio of 27, -18, 12,-8…

-2/3


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 *r296 = a1 *r5


Example


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:

Diverging – Has NO Sum


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


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