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8.3 Geometric Sequences

DEFINITION OF A GEOMETRIC SEQUENCEA geometric sequence is a sequence of the form

The number a is the first term, and r is the common ratio of the sequence. the nth term of a geometric sequence is given by

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Write the first 5 terms and the nth term of the geometric sequence if a = 3 and r = 2

Find r, a and the nth term of the following sequence:

r =      ,   a = 1,   an = 1(  )n ­ 113

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Are the following sequences geometric? If so, find r and the nth term.

2,­10, 50, ­250, 1250,...

3, 6, 9, 12,...

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an = 4 + 3n an = (­1)n 2n

an = nn

Are the following sequences geometric? If so, state the first 5 terms, the common ratio, & express the nth term in the form an = a(r)n - 1.

terms:

Not geometric

GeometricNot geometric

terms:terms:

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Find the 8th term of the geometric sequence5, 15, 45,...

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The third term of a geometric sequence is , and the sixth term is . Find the fifth term.

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The second and fifth terms of a geometric sequence are 30 & 3750, respectively. Which term of the sequence is 468,750?

468,750 is the 8th term.

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PARTIAL SUM OF A GEOMETRIC SEQUENCE

For the geometric sequence an=arn­1, the nth partial sum

is given by

This is also called the sum of a finite geometric series.

(proof on page 610 in textbook)

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Find the sum of the first five terms of the geometric sequence

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Find the sums:­1

38581

­770243

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Using a picture and partial sums find

Partial sums: 164+

132+

14+

12 1

8+

116+

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if

then

We use this to find the actual sum of

If |r|<1, then the value of this term goes to zero as n goes to infinity.

So this is the formula for an infinite geometric series if |r|<1.

Example:

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SUM OF AN INFINITE GEOMETRIC SERIESIf |r|<1, then the infinite geometric series

converges and has a sum

If |r|>1, the series diverges.

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Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find the sum.

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Find the fraction that represents the rational numbers: 0.123123123...

0.123123123...

First write the repeating decimal as a sum:

Next write each decimal as a fraction:

Now plug "a" and "r" into the formula:

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Assignment:

Section 8.3 on Webassign