This document provides information about geometric sequences. It defines a geometric sequence as a sequence where each term is obtained by multiplying the preceding term by a common ratio. The key points are: - A geometric sequence is defined by a common ratio r that is used to multiply the preceding term to obtain the next term. - To find missing terms, the common ratio r is used to multiply or divide the preceding term. - The formula for the nth term an of a geometric sequence is an = a1rn-1, where a1 is the first term and r is the common ratio. - Examples are provided to demonstrate finding missing terms and calculating the nth term using the formula. - The

1. The 2nd type of Sequence
GEOMETRIC
SEQUENCE

2. OBJECTIVES:
At the end of this session, you will be able to:
• solve inductively the difference between arithmetic sequence and
geometric sequence using real-life situations.
• determine a geometric sequence;
• identify the common ratio of a geometric sequence;
• find the missing term of a geometric sequence;
• determine whether a sequence is geometric or arithmetic; and
• find the nth term of a geometric sequence;

3. GEOMETRIC
SEQUENCE

4. sequence obtained by multiplying a common rat
the preceding terms in order to get the succeedi
terms.

5. Find the common ratio and the next three terms of each geometric
sequence.
3, 9, 27, 81 …
Next 3 terms
3 243, 729, 2187
Common Ratio(r)

6. Find the common ratio and the next three terms of each geometric
sequence.
7, 14, 28, …
Next 3 terms
2 56, 112, 224
Common Ratio(r)

7. Find the common ratio and the next three terms of each geometric
sequence.
-256,-64,-16, …
Next 3 terms
𝟏
𝟒
-4, -1, -
𝟏
𝟒
Common Ratio(r)

8. Finding the missing
term/s of a geometric
sequence

9. Remember:
To identify the missing term:
1. Find the common ratio
2. If the unknown value is a succeeding
term, then multiply the preceding term
to the common ratio.
3. If the unknown value is a preceding term,
then divide the succeeding term by the
common ratio.

10. Identify the value of the missing term of the
following geometric sequences:
1. 1, 3,12, 48, _____
- The missing term is a succeeding term and comes after 48.
- The common ratio is 4
- To obtain the missing term, multiply 48 to 4
- Therefore, the missing term in the geometric sequence is 192.
48 x 4 = 192
2. ____, 32, 64, 128
- The missing term is a preceding term and comes before 32.
- The common ratio is 2
- To obtain the missing term, divide 32 by 2.
- Therefore, the missing term in the geometric sequence is 16
32 ÷ 2 = 16

11. 27, __, 3, 1, 1/3
9
___6, 12, 24, 48, ...
3

12. Finding the missing
term/s of a geometric
sequence

13. The pattern in
the table shows
that to get the
nth term,
multiply the first
term by the
common ratio
raised to the
power n – 1.
To find the output an of a geometric sequence
when n is a large number, you need an equation,
or function rule.

14. If the first term of a geometric sequence is a1,
the nth term is an , and the common ratio is r,
then
an = a1rn–1
nth term 1st term Common ratio

15. Example 1:Finding the nth Term of a Geometric
Sequence
The first term of a geometric sequence is 500,
and the common ratio is 0.2. What is the 7th
term of the sequence?
an = a1rn–1
Write the formula.
a7 = 500(0.2)7–1
Substitute 500 for a1,7 for n, and
0.2 for r.
= 500(0.2)6 Simplify the exponent.
= 0.032 Use a calculator.
The 7th term of the sequence is 0.032.

16. Example 2: Finding the nth Term of a Geometric
Sequence
For a geometric sequence, a1 = 5, and r = 2.
Find the 6th term of the sequence.
an = a1rn–1
Write the formula.
a6 = 5(2)6–1
Substitute 5 for a1,6 for n, and 2
for r.
= 5(2)5 Simplify the exponent.
= 160
The 6th term of the sequence is 160.

17. Example 3: Finding the nth Term of a Geometric
Sequence
What is the 9th term of the geometric
sequence 2, –6, 18, –54, …?
2 –6 18 –54
an = a1rn–1 Write the formula.
a9 = 2(–3)9–1 Substitute 2 for a1,9 for n, and –3
for r.
= 2(–3)8 Simplify the exponent.
= 13,122 Use a calculator.
The 9th term of the sequence is 13,122.
The value of r is
–3.

18. When writing a function rule for a sequence with
a negative common ratio, remember to enclose r
in parentheses. –212 ≠ (–2)12
Caution

19. Example 4
What is the 8th term of the
sequence 1000, 500, 250,
125, …?

20. Example 4
What is the 8th term of the sequence 1000,
500, 250, 125, …?
1000 500 250 125
an = a1rn–1 Write the formula.
Simplify the exponent.
= 7.8125 Use a calculator.
The 8th term of the sequence is 7.8125.
The value of r is .
Substitute 1000 for a1,8 for n, and
for r.
a8 = 1000( )8–1

21. Arithmetic sequences
vs. geometric
sequences

22. 5, 12, 19, 26, 33,…
An arithmetic sequence is a
sequence obtained by adding a
common difference (d) to the
preceding terms in order to obtain
the next terms.
3 , 12, 48, 192…
A geometric sequence is a
sequence obtained by multiplying a
common ratio(r) to the preceding
terms in order to obtain the
succeeding terms.

23. ARITHMETIC OR GEOMETRIC
1. 12, 15, 18, 21,…
1. 10, 5, 0, -5,…
2. 5, 15, 45, 135,…
4. -24, 12, -6, 3,…

24. What have you learned today?
What is geometric sequence?
How will you get the next terms of a geometric sequence?
What is the difference between arithmetic sequence and geometric
sequene?

25. Give an example that
illustrates geometric
sequence in real-life
situations.

26. Am I Arithmetic or Geometric?
ACTIVITY:
Determine whether the sequence is geometric or arithmetic. If it is geometric
sequence write the common ratio(r), and if it is arithmetic write the
common difference(d).

27. 1. 12, 15, 18, 21,…
2. 10, 5, 0, -5, …
3. 5, 15, 45, 135, …
4. -24, 12, -6, 3, …

28. 1. 12, 15, 18, 21,…
2. 10, 5, 0, -5, …
3. 5, 15, 45, 135, …
4. -24, 12, -6, 3, …
d=3
ARITHMETIC SEQUENCE
ARITHMETIC SEQUENCE d=-5
GEOMETRIC SEQUENCE r=3
GEOMETRIC SEQUENCE r=-1/2

29. Outputs/Tasks
Q1 Week 6 – October 20, 2021
1. Seatwork n0. 4 (10 points)
2. First Quarter Performance Task: My Digital
Portfolio and Notebook lesson
(Due: November 3, 2021)
3. Finish all tasks in Google classroom
- 4 Written Works
- 2 Google forms assessment
- Arithmetic sequence: Additional Act.
THANK YOU SO MUCH MY DEAR STUDENTS!
Deduction
points for
late
submission:
1 point per
day after the
due date of
submission.