Pre – Calculus
Chapter 10
Section 10.3 Notes: Geometric Sequences and Series
Objectives:
- Find nth terms and geometric means of geometric sequences
- Find sum of n terms of geometric series and the sums of infinite geometric series
DEFINITIONS
Geometric Sequence: when the ratio between consecutive terms is a constant.
Common Ratio: denoted r, the constant ratio between terms.
*To find r, divide any term by its previous term.
*To find the next term in the sequence, multiply the given term by the common ratio.
Example 1: Determine the common ratio and find the next three terms of the geometric sequence.
a) –6, 9, –13.5, … .
b) 243n – 729, –81n + 243, 27n – 81, … .
c) 24, 84, 294, … .
d) 8, -2, , … .
1
2
EQUATION
Example 2: Write an explicit formula for finding the nth term in the geometric sequence.
a) –1, 2, –4, … .
b) 3, 16.5, 90.75, … .
Example 3:
a) Find the 11th term of the geometric sequence –122, 115.9, –110.105, … .
b) Find the 25th term of the geometric sequence 324, 291.6, 262.44, ….
c) Find r when a2 = -18 and a5 =
2
3
d) Find r when a2 = -4 and a6 =-1024
The graphs of terms of a geometric sequence lie on a curve, as shown.
A geometric sequence can be modeled by an exponential function
in which the domain is restricted to natural numbers.
Example 4:
a) A couple purchased a home for $225,000. At the end of each year, the value of the home appreciates 3%. Write an explicit formula
for finding the value of the home after n years.
b) A couple purchased a home for $225,000. At the end of each year, the value of the home appreciates 3%. What is the value of the
home after the tenth year?
GEOMETRIC MEAN
•
•
If two nonconsecutive terms of a geometric sequence are known, the terms can be calculated.
General steps to do so:
• Find the common ratio using the nth term formula for geometric sequences.
• Use r to find the geometric means (next terms)
Example 5:
a) Write a sequence that has three geometric means between 264 and 1.03125.
b) Write a geometric sequence that has two geometric means between 20 and 8.4375.
- Part 2 –
DEFINITIONS
Geometric Series: the sum of the terms of a geometric sequence.
Example 6:
a) Find the sum of the first eleven terms of the geometric series 4, –6, 9, … .
b) Find the sum of the first n terms of a geometric series with a1 = –4, an = –65,536, and r = 2.
c) Find the sum of the first 8 terms of the geometric series 8 + 36 + 162 + … .
Example 7: Find
a)
b)
INFINITE GEOMETRIC SERIES
Recall, that an infinite arithmetic series exists when the sequence converges.
An infinite geometric series will exist if |r| < 1
Example 8: If possible, find the sum of the infinite geometric series.
a) 24 + 18 + 13.5 + … .
b) 0.33 + 0.66 + 1.32 + … .
c)
d)
6
12( )𝑛−1
5
Example 9: Express the series using sigma notation. Then find the indicated sum.
a) 3 + 12 + 48 + … + 3072
b) 0.2 – 1 + 5 - … - 625
Example 10:
a) In a geometric sequence a2 = –8 and a7 = 8192 Find S10.
b) In a geometric sequence a3 = 48 and a8 = 1536 Find S8.
c)
1
20
+
1
40
+
1
80
+⋯