Geometric Sequences & Series 8.3

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

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

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

Geometric Sequences & Series 8.3