What is it
The
nth term of a geometric sequence is an = a1r
n-1
Example 1: Find the
tenth term of the geometric sequence 1, -4, 16, -64, …
Solution: a1 = 1; r = -4; n = 10
an = a1
r n-1
a10 = 1(-4)10-1 Substitute
with the given information
a10 = 1(-4)9 Simplify the exponent
a10 = 1(-262, 144) Simplify (-4
a10 = -262, 144 Simplify
Therefore the 10th term is -262, 144.
Example 2: A certain substance decomposes and loses
20% of its weight each hour. If the
original quantity of the substance is 500 grams, how much remains after 8
hours?
Solution: In
this geometric sequence, a1 = 500 and r = 0.80 (100% -
20% = 80% or 0.80). We are asked to find
the amount remaining after 8 hours.
Thus, we must find the amount remaining at the beginning of the 9th
hour.
a1
= 500, r =
0.80, n
= 9
an = a1r
n-1
a9 =
(500)(0.80)9-1 Substitute with the given
information
a9
= (500)(0.80)8
a9 =
83.88 grams Simplify using PEMDAS rule
What’s More
Answer the following:
1) Find the
7th term of the geometric sequence with a1 = 3 and r =
5
2) A machine
costing P1,000,000 depreciates in value 20 percent each year. How much will it be worth at the end of 4
years?
What is it
|
A) Geometric means are the terms between any two
nonconsecutive terms in a geometric sequence.
If the numbers a1, a2, a3….,
an form a geometric sequence, the a2,
a3…., an-1 are called geometric means
between a1 and an. Thus,
the formula to be used is an = a1 r n-1 B) Let m be the geometric mean between two numbers a
and b, so that a, m and b
form a geometric sequence, so that Example 1: Insert three geometric means between 5 and
3125. Solution: Let a1 = 5 and a5
= 3125. We will insert a2,
a3, and a4. a5 = a1r
n-1 3125 = 5r 5-1 Substitute 3125 = 5r4 Simplify the exponent 625 = r 4 Divide by 5 on both sides 54 = r 4 Rewrite the base
625 into a power of 4 (625= 5 r =
If r = 5, the geometric
means are: a2 = 5(5) = 25, a3 = 5(25) = 125, a4
= 5(125) = 625 Thus, the sequence is 5, 25,
125, 625, 3125.
If r = -5 then the
geometric means are: a2= -5(5) = -25, a3 = -5(-25) =125, a4
= -5(125) = -625 Thus, the sequence is 5, -25,
125, -625, 3125.
Example 2: Find the geometric mean between 4 and
36. Solution: a
= 4, b = 36 m
= m
= m
= 12 Extract the root Therefore, the geometric mean
between 4 and 36 is Note: The geometric mean between
two numbers a and b is -
|
|
|
|
Answer the following.
1) Insert three geometric means between 16 and
1296.
2) Find the geometric mean between 10 and
|
Lesson
3: Sum of a Finite and Infinite Geometric Sequence. |
What is it
Finite
sequence if it has a first term and a last term.
The formula for the sum of the first n-terms in a
geometric sequence is 
Where: Sn = sum
a1 = the first term
r = the common ratio, r .
Example
1: Find the sum of the first 5 terms
of 3, 6, 12, 24, …
Solution: n = 5 ; a1 = 3; r = 2
Sn =
S5 =
S5 =
S5
= 93 Therefore
the sum of the first 5 terms of 3, 6, 12, 24, … is 93.
Example
2: Find the sum of the sequence -5, -10,
-20, -40, … until a6?
Solution: n
= 6 a1 =
-5; r = -2
Sn
=
S5
=
S5 =
S5 = -315 Therefore
the sum is -315
The
sum S of the terms of an infinite geometric sequence is
where a1
= the first term
r = the common ratio,
and / r / < 1.
Example
1: Find the sum of 64+ 32+16+ …
Solution:
a1 =
64; r = ½
S∞ =
S∞=
S∞ =
S∞ = 128 Therefore
the sum is 128.
Example
2: Find the sum of
Solution:
What’s More
Solve the following.
1) The sum of the first 5 terms of the geometric
sequence 4, 12, 36, 108, …
2) The sum of the infinite geometric sequence 64, 16, 4, 1, …
Assessment
Directions: Read and
understand the problems carefully. Write your answer on the answer sheet
provided for you. STRICTLY NO ERASURE.
1. Which of the following describes a finite
geometric sequence.
A.
It has infinite number of terms C.
The sum of terms does not exist sometimes
B.
It has n terms D. It has set of even integers
2. Find the geometric mean of 3 and 48.
A.
6 B. 12 C.
24 D.
26
3. Find the sum of the infinite geometric
sequence
A.
-1 B.
4. Find the sum of the first 10 terms of the
geometric sequence 3, -6, 12, -24, …
A. -3609 B. -1024 C. -1023 D. 3609
5. Find the 7th term of the geometric
sequence 1, 3, 9, …
A. 729 B. 2187 C. 1093 D. 3280
6. The sum of the first 11 terms of the
progression 2, -2, 2, -2 ….is
A. -2 B. 0 C. 2 D. 22
7. If the series has an infinite number of
terms, then it is _________.
A. an infinite sequence C. a finite
series
B. a series D. a function
8. The geometric mean of two numbers is equal to
the ______________.
A. square root of the product of the two numbers
B. difference of the two numbers
C. product of the two numbers
D. quotient of the two numbers
9. If each bacterium divides into 4 bacteria
every hour, how many bacteria will be present at the end of 5 hours if there
are 4 bacteria at the start?
A.
64 B. 256 C. 1024 D. 4096
10. Find the geometric mean between
0 comments:
Post a Comment