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Lesson 1: Geometric Sequence |
Study the given
sequences and see if you can get the pattern or rule.
1) 2,
4, 8, 16, …
2) -1,
5, -25, 125, …
3) 80,
40, 20, 10, 5, …
You need the concept of ratio in order to
understand the next kind of sequence.
Example 1: Given the sequence 2, 4, 8, 16
8 is the product of 4 and 2
16 is the product of 8 and 2
Since the succeeding term is the product of the
previous term and a constant 2 (ratio), therefore 2, 4, 8, 16 is a geometric
sequence.
Example 2: Given:
1, 4, 9, 16, …
4 is the product of 2 and 2
9 is the product of 3 and 3
16 is the product of 4 and 4
Since the succeeding term is NOT a product of the previous term and a
constant (ratio), therefore the above sequence is NOT a geometric
sequence.
NOTE: The common
ratio is found by dividing after the first by the preceding term. For example in the sequence 2, 4, 8, 16
Identify
whether Geometric or NOT.
1) 2,
-6, 18, -54, …
2) 1,
8, 27, 64, …
3) 80x,
40x, 20x, 10x, …
4)
Lesson 2: The Common Ratio
What is it
Geometric sequence is a sequence in which a term is
obtained by multiplying the preceding term by a constant number called common
ratio r. The general term of
a geometric sequence is,
an
= a1r n-1
where: n = number of terms
a1
= first term
an = nth term
r = common ratio
Example 1:
Find r for the geometric sequence with a1 = 24
and a4 = 648.
Solution: Using the formula,
an
= a1r n-1
648 = 24r 4-1 Substitute with the given
information
27 = r3 Divide by 24 on both
sides and simplify the exponent
33
= r3 Rewrite
the base 27 into a power of 3
(27 = 3
r = 3 Property of equality (If
exponents are equal, then bases are equal)
Example 2: Find r in a geometric sequence if the
first term is -3 and the 6th term is 96.
Solution: Let a1
= -3 n = 6 a6 = 96
an = a1r n-1
96 =
-3r 6-1 Substitute
-32 = r5
Divide by -3 on both
sides and simplify the exponent
(-2)5 = r5 Rewrite the base -32 into a power of 5 (-32 = (-2
r = -2 Property of equality
What’s More
Find the common ratio of the following geometric
sequence.
1) a1
= 2 and a4 = 128
2) a2
= -2 and a5 = -16
Lesson 3: The
Difference Between Geometric Sequence and Arithmetic Sequence
What is it
Like
the arithmetic sequence, each of the terms in a geometric sequence is related
to the preceding term through a definite pattern.
Example A.) 6x, 18x, 54x, 162x…
Answer: Geometric Sequence
Common
ratio: 3
Example B.)
-4, -10, -16, -22…
Answer:
Arithmetic Sequence
Common difference: -6
Example C.)
-60, 30, -15, 7.5, …
Answer:
Geometric Sequence
Common ratio: -
Example D.)
1,
Answer:
Arithmetic Sequence
Common difference:
What’s More
Determine whether each sequence is arithmetic,
geometric or neither. If the sequence is arithmetic, give the common
difference; if geometric, give the common ratio.
1) 1, 5, 9,
13, 17, … 5) 5x2, 5x4, 5x6,
5x8, …
2) -3, 2, 7,
12, … 6) 625, 125, 25, 5, …
3) 5, -10,
20, -40, … 7)
4) 1, 0.6,
0.36, 0.216, …
Assessment
Directions: Read and understand the problems carefully. Write your
answer on the answer sheet provided
for you. STRICTLY NO ERASURE.
1. A geometric
sequence is characterized by a constant _________.
A. ratio B. difference C. number of items D. sum
2. The following
sequences are arithmetic EXCEPT ONE which is a geometric.
A. 2, 8, 14, 20 B.
2, -3, -8, -13 C.
-5, 5, -5, 5 D.
3. Which of the
following is the ratio of a geometric sequence 3, 6, 12, 24?
A.
4. Which of the following is NOT a geometric sequence?
A. 3, 9, 27, 81 B. 9, 3, 1,
5. Find the ratio of a
geometric sequence with a2 = 24 and a5 =
648.
A. 3 B.
-3 C. 4 D. -4
6. Which of the
following is the step in finding the ratio of a geometric sequence.
A. Adding after the first by the preceding term
B. Dividing after the first by the
preceding term
C. Multiplying after the first by
the preceding term
D. Subtracting after the first by the
preceding term
7. The following are
steps in finding the ratio of a geometric sequence 1, 3, 9, 27, EXCEPT one.
A.
8. Identify the
sequence which is geometric.
A. 1, 1, -1, 1 B. 3, 6, 9, 12 C. 2, 4, 6, 8 D. -2, 4, -6, 12
9. Which of these
statements is TRUE.
A. The defining equation of an arithmetic
sequence is quadratic function
B. Each term of a geometric sequence is obtained
by multiplying the preceding term by a common multiplier
C. The terms of an arithmetic sequence has a
common factor
D. A geometric sequence has a common difference
10. Which of these is
a ratio of 7, -7, 7, -7?
A. -
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