Download 12.3 Geometric Sequences Series

Precalculus 12.3 Geometric Sequences; Geometric Series
Objective: able to determine if a sequence is geometric; find a formula for a geometric sequence; find the sum of a
geometric sequence (a geometric series).
A sequence is called a geometric sequence when the ratio between any two successive terms is always the
same number. Thus, a geometric sequence may be defined recursively as
a1 = a, a n = ra n −1 where a1 = a
nonzero number
and
r ≠ 0 are real numbers. The number
ar ,
, or as
a is the first term, and the
r is the common ratio.
The terms of a geometric sequence with first term
a,
a1 = a, a n ÷ a n −1 = r
2
ar ,
a and common ratio r follows the pattern:
3
ar , …
 5  n 
1. Show that the sequence  2   is geometric. Find the first term and the common ratio.


a is the first term of a geometric sequence whose common ratio is r ≠ 0
a
formula for the nth term, n .
Suppose that
. Let’s find a
{a n } whose first term is a and whose common ratio is r, the n
For a geometric sequence
determined by the formula:
th
term is
2. Find the 15th term and the nth of the geometric sequence with initial term of -2, and common ratio of 4.
3. Find the 18th term of the geometric sequence 1, 3, 9, …
Develop a formula for the sum of the first
For a geometric sequence
of the first
n terms of a geometric sequence.
{a n } whose first term is a and whose common ratio is
r , r ≠ 0, r ≠ 1 , the sum S
n
n terms is called a geometric series and is determined by the formula:
4. Find the sum
∑
n
4 ⋅ 3 k −1 .
k =1
{
5. Find the sum of the first 15 terms of 4 ⋅ 3
n −1
}.
An infinite sum of the form a + ar + ar 2 + ar 3 + ⋯ + ar n −1 + ⋯ with first term
infinite geometric series and is denoted by
∞
∑
a and common ratio r, is called an
ar k −1 .
k =1
Find the sum of an infinite geometric series.
If r < 1 , the sum of the infinite geometric series
∞
∑
6. Find the sum of the infinite geometric series 2 +
ar k −1 is
k =1
4 8
+ +⋯ .
3 9
∑
∞
ar k −1 =
k =1
a
.
1− r
7. A ball is dropped from a height of 30 feet. Each time it strikes the ground, it bounces up to 0.8 of the
previous height.
a. What height will the ball bounce up to after it strikes the ground the 3rd time?
b. What is its height after it strikes the ground the nth time?
c. How many times does the ball need to strike the ground before its height is less than 6 inches
d. What total distance does the ball travel before it stops bouncing?
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