GEOMETRIC-SEQUENCE.pptx

Prayer
Dear Lord,
You hold the full of creation in your hands, from the huge and awe-inspiring
universe, to every little grain of sand. You're the creator of all time, you balance night
and day, 𝒊𝒏𝒇𝒊𝒏𝒊𝒕𝒆 and safe.
We ask for your guidance, with your 𝒂𝒓𝒓𝒐𝒘𝒔 𝒐𝒇 𝒔𝒖𝒄𝒄𝒆𝒔𝒔, so that we will 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕𝒍𝒚
have a 𝒓𝒂𝒕𝒊𝒐𝒏𝒂𝒍 and 𝒑𝒐𝒔𝒊𝒕𝒊𝒗𝒆 attitude in everything we do.
Give us the 𝒑𝒐𝒘𝒆𝒓 to overcome challenges and 𝒓𝒐𝒐𝒕 out the bad things in our
hearts.
All these we pray in recognition of your power and love,
Amen.

Drill: Choose the correct answer.
1.What is the value of 4³?
A.12
B.16
C.64
D.81 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
C. 64

2. Which is the lowest term of 12/48?
A.1/4
B.4/5
C.2/3
D.3/4 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A. 1/4

3. What is the quotient when 108 is
divided by 36?
A.3
B.4
C.5
D.6 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A. 3
A. 3

4. Simplify: 2(3)⁴
A.98
B.120
C.128
D.162 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
D. 162

5. What is the value of (3/4)³?
A.27/64
B.9/16
C.27/64
D.9/25 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
C. 27/64

Objectives:
1. Determine a geometric sequence
2. Identify the common ratio of a geometric
sequence
3. Find the missing term
4. Determine whether a sequence is geometric or
arithmetic.

ONSIDER AND ANALYZE THE PROBLEM BELOW.
Acki has purchased 15 books. The prices of
the books are as follows:
Php 20, Php 40, Php 80,...
a. Is there a pattern in the price of the books? If
there is, describe it.
b. What do you think is the price of the next book?

The sequence
20, 40, 80,...
is an example of geometric
sequence.

GEOMETRIC SEQUENCE
Is a sequence where each term after
the first is obtained by multiplying
the preceding term by the same
number called the common ratio (r =
common ratio)

A sequence is geometric if their
ratios are equal. The common ratio r,
can be determined by dividing any
term in the sequence by the term
that precedes it.

Example 1
Determine whether the sequence
4, 12, 36, 108, ... is geometric
Show that their ratios are equal
The common ratio is 3 (r = 3)
Since the ratios are equal therefore the given sequence is a GEOMETRIC.
4, 12, 36, 108,…
a₁ a₂ a₃ a₄
r = a₂/a₁ = a₃/a₂ = a₄/a₄
3 = 3 = 3
12/4 = 36/12 = 108/36

Example 2
Is -72, 36, -16,10, ... a geometric sequence or not?
SOLVE FOR r:
NOT geometric because their ratios are not equal.
36/-72 = - 1/2
-16/36 = -4/9
10/-16 = -5/8

Example 3
Is -128, 64, -32,16,... form a geometric
sequence or not?
SOLVE FOR r:
GEOMETRIC sequence because the values of r are equal.
64/-128 = -1/2
-32/64 = -1/2
16/-32 = -1/2

Any questions how to determine
whether the given sequence is
geometric or not and how to
determine the common ratio?

Suppose you're asked to determine
the next 2 or 3 terms of a geometric
sequence, what will you do to solve
for these?

Multiply the previous term by
r (common ratio) to get the next term.

Example 1
Supply the next 3 terms of the geometric sequence
4, 12, 36, 108, _____, _____, _____
Since r = 3 (refer to example 1),
to solve for the next 3 terms just multiply
So the next 3 terms of the sequence are 324, 972 and 2916.
108(3)=324
324(3)=972
972(3)=2916

Example 2
What are the next 2 terms of the geometric sequence
-128, 64, -32, 16,_____, _____
Since r=-1/2 (refer to example 3),
to solve for the next 2 terms multiply
Therefore the next 2 terms of the geometric sequence are -8
and 4.
16(-1/2)= -8
-8 (-1/2)= 4

Suppose you're ask to determine the
8th, 20th or nth term of a geometric
sequence, what formula are you going
to use?

FORMULA TO SOLVE FOR THE nth
TERM OF GEOMETRIC SEQUENCE
where:
aₙ = indicated term
a₁ = first term
r = common ratio
n = number of terms
aₙ = a₁rⁿ⁻¹

Example 1
Find the 12th term of the geometric sequence
4, 12, 36, 108, ....a₁₂
GIVEN:
r = 3
a₁= 4
n = 12
a₁₂ =?
Substitute the given using the formula
aₙ = a₁rⁿ⁻¹
a₁₂=4(3¹²⁻¹)
a₁₂=4(3¹¹)
a₁₂=4(177,147)
a₁₂= 708,588

Example 2
Solve for the 8th term of the geometric sequence
-128, 64, -32, 16,...a₈
GIVEN:
r = -1/2
a₁= -128
n = 8
a₈ =?
Substitute the given using the formula
aₙ = a₁r ⁿ⁻¹
a₈= -128[ (-1/2)⁸⁻¹]
a₈= -128(⁻1/2)⁷
a₈= -128/-128
a₈= 1

What is the difference between
arithmetic sequence and geometric
sequence?

In ARITHMETIC SEQUENCE the next term is
obtained by adding a constant number called
the common difference (d) while in
GEOMETRIC SEQUENCE the term is obtained
by multiplying a constant number called the
common ratio(r).

A piece of spoiled meat has some bacteria
in it. The number of bacteria increases five
times every hour. If the number of bacteria
is 1000 on the first hour, complete the
sequence until five hours. What is the total
number of bacteria at the end of five hours?

Solution:
GIVEN:
r = 5
a₁= 1000
n = 5
a5 =?
aₙ = a₁r ⁿ⁻¹
a5 = 1000[ (5)5⁻¹]
= 1000(5)4
= 1000(625)
a5= 625, 000

Now let's check if you can identify
arithmetic and geometric sequences.
Determine whether each sequence is
arithmetic or geometric and give the
common difference or common ratio.

1
3, 15, 75, 375,...
Geometric sequence
r = 5
15/3=5
75/15=5
375/75=5

2
20, 10, 5, 5/2, ...
Geometric sequence
r =1/2
10/20=1/2
5/10 =1/2
(5/2)/5 =1/2

3
1, 5, 9, 13, ...
Arithmetic sequence
d = 4
5 - 1 = 4
9 - 5 = 4
13 - 9 = 4

4
3, 2.75, 2.5, 2.25, ...
Arithmetic sequence
d = -0.25
2.75 -3 = - 0.25
2.5 - 2.75= - 0.25
2.25 - 2.5 = - 0.25

5
The seating capacity of a movie house
with 30 rows if there are 20 seats in the
first row, 23 seats in the second row, 26 in
the third, and so on.
Arithmetic sequence
d= 3

6
A piece of spoiled meat has some bacteria in it.
The number of bacteria increases five times
every hour. If the number of bacteria is 1000 on
the first hour, complete the sequence until five
hours.
Geometric sequence
r= 5

GEOMETRIC-SEQUENCE.pptx

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