Geometric Sequences
It is a sequence of numbers where each term after the first is found by multiplying the previous one by a
fixed, non-zero number called the common ratio. It is a sequence where each term is found by
multiplying or dividing the same value from one term to the next.
GENERATING PATTERN
Example: Find the next three terms of the geometric sequence 3, 21, 147, …
Solution: To find the next three terms of the sequence:
First, identify the common ratio ( r ).
Afterwards, multiply the obtained common ratio ( r ) to the preceding term to get the next term.
a1 = 3 a4 = 147 x 7 = 1 029
a2 = 3 x 7 = 21 a5 = 1 029 x 7 = 7 203
a3 = 21 x 7 = 147 a6 = 7 203 x7 = 50 421
Therefore, the next three terms of the geometric sequence are 1 029, 7 203, and 50 421.
Identify the common ratio and the next 2 term of the sequence. 80, 20, 5,…
Solution:
Geometric Sequences VS Arithmetic Sequences
A sequence is Arithmetic if there is a constant number, the common difference, d, that is being
added or subtracted to get the next term. And if the sequence is Geometric, there is a constant number,
the common ratio, r, that is being multiplied or divided to get the next term to form a sequence.
Finding the nth Term of a Geometric Sequence
On the previous lesson in this module, you learned about geometric sequences and how to find the next
terms of it. On this lesson, you will learn how to find the nth term of geometric sequences.
Example: Find the 6th and 9th term of the sequence 2,4,8,1
Find the 8th and 10th term of the sequence 6, 12,24,48, …
Geometric Mean
Example 1: Find the positive geometric mean of 4 and 16.
Solution: GM= √(4∙16) = √64 =8 Therefore, the geometric mean between 4 and 16 is 8.
Example 2: Insert 3 geometric means between 2 and 162.
Solution: 2, ¬__, ___, ___, 162
First, we will find the geometric mean between 162 and 2.
√(162∙2) = √324 = 18
Now, we have 2, ¬__, 18 , ___, 162
Next, we will look for the geometric mean between 2 and 18.
√(18∙2) = √36 = 6
The ratio is 3 since 6/2=3. We can now complete the sequence as: 2, ¬6, 18 , 54, 162.
Find the geometric mean of 12 and 108.
Insert the three geometric means between 5 and 1 280.
SUM OF A GEOMETRIC SERIES
Finding the sum of Finite and Infinite Geometric Sequence
Solve Problems Involving Sequences
Arithmetic Sequence Sample Problem
A company offers Jeffrey a starting annual salary of P240,000 and a yearly increase of P20,000
after the first year. What will be his annual salary on the 10th year?
Given a1 = 240 000 n = 10 d = 20,000
Solution: an = a1 + (n-1)d
a10 = 240,000 + (10-1) 20,000
a10 = P420,00
Geometric Sequence Sample Problem
During an initial phase of an outbreak of measles, the number of infections can grow
geometrically. If there were 4, 8, 16, … on the first three days of an outbreak of measles, how many will
be infected on the 6th day?
Given: a1 = 4 n=6 r=2
Solution: By substituting the given values in the formula: an = a1r n-1
a6 = 4(2)5
a6 = 128 Therefore, a6 = 128
Harmonic and Fibonacci Sample Problem
Given the arithmetic sequence -20, -26, -32, -38, …, find the first 8 terms of its corresponding
harmonic sequence.
Solution: Completing the 8 terms of the given arithmetic sequence, we have -20, -26, -32, - 38, -44, -50, -
56, and -65.
Therefore, the first 8 terms of the harmonic sequence are -1/20, -1/26, -1/32, -1/38, -1/44, -1/50, -
1/56and -1/62.
Given the Fibonacci sequence 5, 8, 13, 21, 34, …, find the next 6 terms.
Solution: Each term in a Fibonacci sequence can be obtained by adding its two consecutive preceding
terms, hence a6 = 21 + 34 = 55 a9 = 89 + 144 = 233
a7 = 34 + 55 = 89 a10 = 144 + 233 = 377
a8 = 55 + 89 = 144 a11 = 233 + 377 = 610
Thus, the next 6 terms are 55, 89, 144, 233, 377, and 610.
Part 1
I. Identify each sequence as arithmetic, geometric or neither.
____________1. 4, 8, 32, 256,… ____________4. 0.25, 0.5, 1, 2,…
____________2. 6, 6.2, 6.4, 6.6,… ____________5. 11, 14, 17, 20,…
____________3. 1.1, 1.01,1.001,…
II. Identify the common ratio and 4th and 5th term of the following sequences.
6. 6, 30, 150,… 7. 1/3, 2/9, 4/27,… 8. 2, -6, 18, -54,…
III. Geometric Mean
9. Find the geometric mean of 64 and 4. 11. Find the geometric mean between 16 and
25.
10. Insert 3 geometric means between 64 and 4. 12. Insert 3 geometric means
between 6 and 96.
IV. Find the first three terms of each geometric sequence.
14. a1=2, r=4 15. a1=48, r=-"1" /"3"
V. Sum of Finite and Infinite Geometric Sequence
15. Find the sum of the first 10 terms of the geometric sequence: 3, 12, 48, 192,…
16. Find the sum of the first 6 terms of geometric sequence: 125, 25, 5,…
17. Find the sum of an infinite geometric series whose first term is 4 and common ratio is
"1" /"5" .
18. Given the sequence 9, 3, 1,… , find "S" _"∞" .
19-20. PROBLEM SOLVING
A ball tossed to a height of 4 meters rebound to 40% of its previous height. Find the distance the
ball has traveled when it strikes the ground for the 5th time.
Using Venn diagram, show the common and differences between arithmetic sequence and geometric
sequence.
THEY NEED HELP!
INSTRUCTION: Read and analyze the situation carefully. Then, provide what is being asked.
SITUATION: Your classmates Ruben and Nathaniel are arguing during your Math subject. You found out
that they are arguing about the examples of sequences presented by your teacher.
Ruben is telling that example A is an arithmetic sequence, while example B is a geometric sequence. On
the other hand, Nathaniel is defending that example A is a geometric sequence, while example B is an
arithmetic sequence.
Since you are knowledgeable about the issue, who among your classmates will you agree with?
ANSWER: ____________________________________________________________
____________________________________________________________
What explanation/s will you provide to your classmates for them to understand clearly the difference
between an arithmetic sequence and geometric sequence?
ANSWER:
¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬___________________________________________________________________
___________________________________________________________________
____________________________________________________________________
____________________________________________________________________
References:
Gladys C. Nireva, Ph.D. et. al.2018 Grade 10 Mathematics Patterns and Practicalities revised
edition, Antonio Rnaiz cor. Chino Roces Avenues, Makati, City, Salesiana Books by Don Bosco Press Inc.
https://www.thoughtco.com/how-to-calculate-the-mean-or-average-609546#:~:text=The
%20average%20is%20simply%20the,the%20average%20or%20arithmetic%20mean.
Haydee C. Hitosis et. al.,Mathtek10 Volume1, 101 V. Luna Ave.,Sikatuna Village, Diliman, Quezon, City,
Techfactors INC.
Mathematics – Grade 10, Alternative Delivery Mode,Quarter 1 – Module5: Illustrate a geometric
sequence & Module 6: Differentiates a geometric sequence from an arithmetic sequence. - M10AL-Id-2;
First Edition, 2019: Published by the Department of Education
Orlando A. Oronce & Marilyn O. Mendoza; E-MATH 10; Rex Bookstore
