Notes – Geometric Sequence and Series
H-Algebra 2
Common Ratio, r = ____________________________
Date: _____________________
Geometric Sequence: A geometric sequence is a sequence in which each term is found by multiplying the term preceding
it by a constant. This constant is called the common ratio, and is represented by the letter r.
Examples of Geometric Sequence:
2, 4, 8, 16, 32, 64, …; r = 2
5, 5, 5, 5, 5, 5, …; r = 1
100, -20, 4, -0.8, 0.16, …; r = -1/5
Non-Examples of Arithmetic Sequence:
2, 4, 6, 8, 16, … (adds 2, not multiplies by 2)
1, 2, 4, 8, 32, 64, … (number to multiply not always 2)
Why is it called a common ratio? Doesn’t ratio mean comparison? Or fraction?
Ratio does mean comparison/fraction. In order to find the common ratio in a geometric sequence, take a term and divide
it by the term before it. In the first example, r = 8 / 4 = 2.
We will graph a geometric sequence to see if we can find any similarities with continuous functions.
Know this: instead of using x and y - or f(x) - as our independent and dependent variables, respectively, we use n and an.
n
an
1
0.25
2
0.5
3
1
4
2
5
4
6
8
7
16
Find the common ratio, r:
What type of continuous function does this resemble? (Think back!)
Why? What about this graph gives evidence to this shape?
Geometric Sequence:
a n  a1  r n 1
an = the nth term. EX: a5 is the 5th term. It takes the place of y or f(x).
a1 = the first term in the sequence.
n = the number of a term (NOT its value). For the 5th term, n = 5. It takes the place of x.
r = the common ratio, found by dividing any term by the one before it.
*** Note: when you plug all constants in, the expression can sometimes be simplified. ***
EX #1: Write a simplified form of the geometric sequence with a1 = 8 and r = 3.
EX #2: Write a simplified form of the geometric sequence shown below.
3, 6, 12, 24, 48, …
EX #3: Write each of the sequences from Examples 1 and 2 using a recursive rule.
ex 1:
ex 2:
EX #4: Given geometric sequence with r = -1/4, what is a32 if a29 is -80?
EX #5: Write a simplified form of the geometric sequence with a1 = 128 and a8 = 2187.
EX #6: Write a simplified form of the geometric sequence with a6 = 8 and a13 = 1024.
EX #7: Write a simplified form of the geometric sequence with a4 = 18 and a12 = 118098.
EX #8: Write each of the sequences from Examples 5, 6, and 7 using a recursive rule.
ex 5:
ex 6:
ex 7:
Geometric Series
The derivation of the formula for the sum of a geometric series is more complicated than that of an arithmetic
series, so we will get right to the formula and how to use it.
1 r
S n  a1
1 r
n
Sn = Sum of the first n terms
n = the number of terms in the series
a1 = the first term in the series
r = the common ratio
EX #9: Find the sum of the first 20 terms of the sequence 2, 4, 8, 16, 32, …
EX #10: Find the sum of the geometric sequence with a1 = 100, r = - 1/2, and n = 12
a n  a1  r n 1
1 r n
S n  a1
1 r
EX #11: Find the first term of the geometric sequence with common ratio 5/4 if the fifth partial sum is 1,050.5.
7
EX #12: Evaluate
 23
k 1
k 1
 3
256  

 2
m 1
7
EX #13: Evaluate
m
a n  a1  r n 1
1 r n
S n  a1
1 r
EX #14: Find the sum of the first 8 terms of a geometric sequence with terms a3 = 972 and a7 = 192.
EX #15: The sum of the first n terms of a geometric sequence with common ratio 2 and first term 5 is 5,115.
What is n?