illustrate geometric sequence, finding the nth term of a geometric sequence, finding the first term of the geometric sequence, determining the term representing the given term of a geometric sequence.

1. GEOMETRIC
SEQUENCE
WEEK 3
SEPTEMBER 18 – 22, 2023

2. OBJECTIVES
• Illustrate a geometric sequence
• Differentiate geometric sequence from arithmetic sequence

3. Think-pair-share
Materials: paper and scissors
Procedure:
a) Cut the paper in half
b) Stack the halves. Cut the stack in halves.
c) Continue stacking and cutting the paper strips about half an inch
wide.
Record the total number of pieces obtained in the given table.

4. Number of cuts 1 2 3 4 5 6 7 8
Number of pieces
New number of pieces
𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑛𝑢𝑚𝑏𝑒𝑟𝑜𝑓 𝑝𝑖𝑒𝑐𝑒𝑠
QUESTIONS TO PONDER:
1. WHAT DO YOU OBSERVE ABOUT THE CHANGE IN NUMBER OF PIECES
WITH EACH CUT AFTER THE FIRST?
2. LIST THE NUMBER IN ROW 2 FROM LEAST TPO GREATEST, SEPARATE
THE NUMBERS WITH COMMA.
3. WHAT IS THE COMMON RATIO BETWEEN ANY TWO CONSECUTIVE
TERMS IN THE SEQUENCE?

5. Complete the table

6. GEOMETRIC SEQUENCE
• A SEQUENCE IN WHICH EACH TERM IS OBTAINED BY MULTIPLYING THE
PRECEDING TERM BY A FIXED NUMBER.
• COMMON RATIO (r) IS THE FIXED NUMBER MULTIPLIED IN EACH TERM
OF A GEOMETRIC RATIO.
nth TERM OF A GEOMETRIC SEQUENCE
𝑎𝑛 = 𝑎𝑛−1 ∙ 𝑟 or 𝑎𝑛 = 𝑎1 ∙ 𝑟𝑛−1

7. Example1
• Write a formula for the nth term of the given geometric sequence.
8, 2, ½ , 1/8, 1/32, . . .
Solution: Find the common ratio: r =
𝑎2
𝑎1
=
2
8
=
1
4
𝑎𝑛 = 𝑎1 ∙ 𝑟𝑛−1
𝒂𝒏 = 𝟖 ∙ (
𝟏
𝟒
)𝒏−𝟏
Check: 𝑎1 = 8 ∙
1
4
1−1
= 8 𝑎2 = 8 ∙
1
4
2−1
= 2
𝑎3 = 8 ∙
1
4
3−1
=
1
2
𝑎5 = 8 ∙
1
4
5−1
=
1
32
Note: any number raise to 0 is
equal to 1.

8. Example 2
• WRITE THE FIRST THREE TERMS OF A GEOMETRIC SEQUENCE WHOSE
nth TERM IS GIVEN BY 𝑎𝑛 = 3(−5)𝑛−1.
SOLUTION:
𝑎1 = 3(−5)1−1 𝑎2 = 3(−5)2−1 𝑎3 = 3(−5)3−1
𝑎1 = 3(−5)0
𝑎2 = 3(−5)1
𝑎3 = 3(−5)2
𝑎1 = 3 1 𝑎2 = 3(−5) 𝑎3 = 3(25)
𝑎1 = 3 𝑎2 = −15 𝑎3 = 75
*AFTER GETTING THE FIRST TERM, THE OTHER TERMS MAY BE OBTAINED
BY MULTIPLYING THE PREVIOUS TERM BY -5.

9. EXAMPLE 3
• FIND THE 8th TERM OF THE GEOMETRIC SEQUENCE 24, 12, 6, 3, . . .
GIVEN: 𝑎1 = 24 r =
12
24
=
1
2
𝑎8 = ?
SOLUTION: 𝑎𝑛 = 𝑎1𝑟𝑛−1
𝑎8 = 24(
1
2
)8−1
substitute the values in the formula
𝑎8 = 24(
1
2
)7
𝑎8 = 24(
1
128
)
𝒂𝟖 =
𝟐𝟒
𝟏𝟐𝟖
or 𝒂𝟖 =
𝟑
𝟏𝟔

10. EXAMPLE 4
FIND THE FIRST TERM OF A GEOMETRIC SEQUENCE, IF THE 6th TERM IS 96
and r = 2.
GIVEN: 𝑎6 = 96 r = 2 𝑎1 = ?
SOLUTION: 𝑎6 = 𝑎1(2)6−1
96 = 𝑎1(2)5
(2)5
= 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 = 32
96 = (32)𝑎1
96
32
=
(32)𝑎1
32
𝒂𝟏 = 𝟑
Sequence is
3, 6, 12, 24, 48, 96, . . .

11. EXAMPLE 5
• FIND THE COMMON RATIO OF A GEOMETRIC SEQUENCE, IF THE FIRST
TERM IS 789 AND THE FIFTH TERM IS 12 624.
GIVEN: 𝑎1 = 789 𝑎5 = 12 624 r = ?
SOLUTION: 𝑎5 = 𝑎1𝑟5−1
12 624 = 789(𝑟)4
12 624
789
=
789𝑟4
789
16 = 𝑟4
4
16 =
4
𝑟4
𝟐 = 𝒓

12. EXAMPLE 6
• IN A GEOMETRIC SEQUENCE THE FIRST TERM IS 1 AND THE COMMON
RATIO IS 4, WHAT TERM DOES 1024 REPRESENTS?
SOLUTION: 𝑎𝑛 = 𝑎1𝑟𝑛−1
1 024 = 1(4)𝑛−1
1 024 = (4)𝑛−1
45 = 4𝑛−1
5 = 𝒏 − 𝟏
𝟓 + 𝟏 = 𝒏
6 = 𝒏

13. PRACTICE TASK
• WRITE WHETHER THE GIVEN SEQUENCE IS ARITHMETIC, GEOMETRIC,
OR NEITHER. IF IT IS ARITHMETIC, GIVE THE COMMON DIFFERENCE. IF
IT IS GEOMETRIC, GIVE THE COMMON RATIO
1) 11, 14, 17, 20, . . .
2) 4, 8, 16, 32, . . .
3) 5, 8, 12, 17, 26, . . .
4) 100, -50, 25, -12.5, . . .
5) 1, 8, 27, 64, . . .

14. PRACTICE TASK
•WRITE THE nth TERM FORMULA OF THE FOLLOWING:
1) 4, 8, 16, 32, . . .
2) 100, -50, 25, -12.5, . . .
3) 1, 3, 9, 27, . . .
WRITE THE 6TH TO 10TH TERMS OF THE SEQUENCE
ABOVE.

15. PRACTICE TASK
•WRITE THE nth TERM FORMULA OF THE FOLLOWING:
1) 4, 8, 16, 32, . . .
2) 100, -50, 25, -12.5, . . .
3) 1, 3, 9, 27, . . .
WRITE THE 6TH TO 8TH TERMS OF THE SEQUENCE ABOVE.

16. ASSIGNMENT
• Write whether the sequence is arithmetic or geometric. Write the nth
term of each sequence.
1)2, 8, 32, 128, 512, . . .
2)3, 12, 48, 192, 768, . . .
3)-35, -32, -29, -26, -23, . . .
4)-24, -14, -5, 6, 16, . . .
5)3, -9, 27, -81, 243, . . .

17. •WHAT IS GEOMETRIC SERIES?
•WHAT IS THE FORMULA FOR THE
GEOMETRIC SERIES?
•GIVE 3 OF YOUR OWN EXAMPLES OF
GEOMETRIC SEQUENCE,WRITE THE nth
TERM FORMULA.

19. JOURNAL ENTRY
• HOW DID YOU FIND THE LESSON ABOUT GEOMETRIC SEQUENCE?
• HOW WILL YOU FIND THE VALUE OF n IF THE GIVEN ARE
𝑎1 = 5, 𝑟 = 3, 𝑎𝑛 = 10935
EXPLAIN YOUR SOLUTION.

20. REFERENCES
• E-MATH BOOK 10 BY ORENCE ET.AL.
• MAKATI LMS
• PREPARED BY: CHRISTINE ORQUIA, LPT 1