copyright © 2011 pearson education, inc. geometric sequences and series section 8.3 sequences,...

Copyright © 2011 Pearson Education, Inc.

Geometric Sequences and Series

Section 8.3

Sequences, Series,and Probability

Copyright © 2011 Pearson Education, Inc. Slide 11-3

11.3

A geometric sequence is defined as a sequence in which there is a constant ratio r between consecutive terms or by giving a general formula that will produce such a sequence.

According to the definition, every geometric sequence has the following form:

a, ar, ar2, ar3, ar4,…

Every term (after the first) is a constant multiple of the term preceding it. Of course, that constant multiple is the constant ratio r.

Note that if a = r then an = arn –1 = rrn –1 = rn.

Geometric Sequences


11.3

Definition: Geometric Sequence

A sequence with general term an = arn – 1 is

called a geometric sequence with common ratio r, where r ≠ 1 and r ≠ 0.

Geometric Sequences


11.3

The indicated sum of the terms of a geometric sequence is called a geometric series.

The “trick” to finding the sum of n terms of a geometric series is to change the signs and shift the terms so that most terms “cancel out” when the two equations are added.

This method can also be used to find a general formula for the sum of a geometric series.

Geometric Sequences


11.3

Let Sn represent the sum of the first n terms of the geometric sequence an = arn –1.

Adding Sn and –rSn eliminates most of the terms:

So the sum of a geometric series can be found if we know the first term, the constant ratio, and the number of terms.

12 nn arararaS

r

raS

raSr

ararSS

arararararrS

arararaS

n

n

nn

nnn

nnn

nn

1

1

11

132

12

Geometric Sequences


11.3

Theorem: Sum of a finite Geometric SeriesIf Sn represents the sum of the first n terms of a geometric series with first term a and common ratio r (r ≠ 1), then

.1

)1(

r

raS

n

n

Geometric Sequences


11.3

In general, it can be proved that rn → 0 as n → ∞, provided that |r| < 1, and r n does not get close to 0 as n → ∞.

So, if |r| < 1 and n is large, then Sn is approximately a/(1 – r).

Furthermore, by using more terms in the sum we can get a sum that is arbitrarily close to the number a/(1 – r).

In this sense we say that the sum of all terms of the infinite geometric series is a/(1 – r).

Infinite Geometric Sequences


11.3

Theorem: Sum of an Infinite Geometric SeriesIf a + ar + ar2 + ··· is an infinite geometric series with |r| < 1, then the sum S of all of the terms is given by

We can use the infinity symbol ∞ to indicate the sum of infinitely many terms of an infinite geometric series as follows:

.1 ra

S

1

12

i

iararara

Infinite Geometric Sequences

copyright © 2011 pearson education, inc. geometric sequences and series section 8.3 sequences,...

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