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
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
Are the following sequences geometric? If so, find r and the nth term.
2,10, 50, 250, 1250,...
3, 6, 9, 12,...
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.
Find the 8th term of the geometric sequence5, 15, 45,...
The third term of a geometric sequence is , and the sixth term is . Find the fifth term.
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.
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)
Find the sum of the first five terms of the geometric sequence
Find the sums:1
Using a picture and partial sums find
Partial sums: 164+
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.
SUM OF AN INFINITE GEOMETRIC SERIESIf |r|<1, then the infinite geometric series
converges and has a sum
If |r|>1, the series diverges.
Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find the sum.
Find the fraction that represents the rational numbers: 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:
Section 8.3 on Webassign