Geometric Series and Sequences || Grade 10 First Quarter

Prayer
 Dearest Father,
◦ We thank you for the wide range of opportunities
and blessings You graphed for us. Forgive us for the
times when our doings fail to be parallel and not
even coinciding to Your will. With You as the center
and domain of our activities, it is surely that all our
negative conditions will be inversed. Fill in with
grace the hollow points in the line of our lives. Help
us to be consistent of our actions. May they be
bound inside Your never ending plane of goodness
and be ways to fulfill our function which is to
increase the circumference of the circle of Your love.
◦ Amen.

Objectives of the Lesson
At the end of the session, the students
will be able to:
1. describe geometric sequence
2. differentiate geometric sequence
from arithmetic sequence
3. find the common ratio and nth of the
geometric sequence

Activity No. The Math in Chess
 Let us read the story about the king in
India who loved to play games. Discover
how this story led to the invention of a
very popular game which most students
today find challenging and mentally
engaging. After you have read the story,
answer the following process questions

PROCESS QUESTIONS
 1. Complete the table below:
 2. Find the ratio between two consecutive terms.
 3. What kind of relationship did you find between
the ratios?
 4. How can you find a way to find the number of
grains inside the 20th
square?
First
Square
Second
Square
Third
Square
Fourth
Square
Fifth
Square
Sixth
Square
1

 Hundreds and hundreds ago, there was a King in India
who loved to play games. But he had gotten bored of the
games that were present at the time and wanted a new game
that was much more challenging. He commissioned a poor
mathematician who lived in his kingdom to come up with a
new game.
 After months of struggling with all kinds of ideas, the
mathematician come up with a game of Chaturanga. The
game had two armies each led by a king who commanded
the army to defeat the other by capturing the enemy King. It
was played on a simple 8x8 square board. The king loved this
game so much that he offered to give the poor
mathematician anything he wished for.”I would like one grain
of rice for the first square of the board, two grains for the
second, four grains for the third and so on doubled for each
of the 64 squares of the game board.” said the
mathematician.


◦ Is that all? Why don’t you ask for gold or silver coins
instead of rice grains? asked the king. “The rice is
sufficient for me”, replied the mathematician. The
king ordered his staff to lay down the grains of rice
and soon learned that all the wealth in his kingdom
would not be enough to buy because the amount of
rice was exhausted before the 30th
square was
reached. You have provided me with such a great
game and yet I cannot fulfill your simple wish. You
are indeed a genius!” said the King and offered to
make the mathematician his top most adviser then.
◦ Are you wondering to know exactly how many
grains of rice would be needed on the 64th
square
and what is the number of office grains would be
needed for all 64 squares ? Let us try to discover by
answering the next activity.

◦ Geometric Sequence is a sequence which each
term is obtained by multiplying the preceding
term by a fixed number called the “common
ratio” (r).
r = preceding term divided by previous term
r = a2
a1
Example find the common ratio of the given geometric sequence
3, -9, 27, -81, 243
r = -9 = -3 r = -81 = -3 r = 243 =-3
3 27 -81
What is a GEOMETRIC SEQUENCE?

DIRECTIONS:
 Determine whether the given
sequence is arithmetic, geometric, or
neither. If its is arithmetic, identify
d, if it is geometric identify r.

1. 100, -50, 25,….
Geometric Sequence
r = -
2
1

DIRECTIONS:
neither. If it is arithmetic, identify d,
if it is geometric identify r.

2. , ,
Geometric Sequence
r =
2
1
4
1
8
1
2
1

DIRECTIONS:

3. 4, -12, 36,….
Geometric Sequence r = -3

DIRECTIONS:

4. 6, -6, 6,….
Geometric Sequence r = -1

DIRECTIONS:

5. 20, 30, 36, 42….
neither

DIRECTIONS:

6. 6, 4, 2,….
Arithmetic Sequence d = -2

DIRECTIONS:
 7. 1, , ,….
neither
2
1
3
1

DIRECTIONS:

8. 3, , 2, ….
2
5
2
3
neither

DIRECTIONS:

9. 25, 33, 41, 49….
Arithmetic Sequence d = 8

DIRECTIONS:

10. 12
, 22
, 32
, 42
….
neither

Write a formula for the nth term of the
given geometric sequence
 an = a1 • r (n -1)
1.) 32, 8, 2, , , ,.....
2
1
r =
32
8
Solution
a1= 32
=
an = 32 • (n -1)
4
1
a1 = 32 • (1 - 1)
4
1
a2 = 32 • (2 - 1)
4
1
8
1
32
1
Find the common ratio
4
1
Since each term is of the term preceding it, the nth
term is given by:
4
1
Check = 32
= 8
8
1
an= ?

To Find the nth term of the given
sequence
 an = a1 • r (n -1)
Find the eleventh term of the given geometric sequence.
2.) 2, 1, ,
2
1
4
1
r =
2
1
a1 = 2 a11= ?
n = 11
a11 = 2 • (11 -1)
2
1
a11 = 2 • (10)
2
1
a11 = 2 •
1024
1
1
512
a11 =
512
1

To Find the nth term of the given
sequence
 an = a1 • r (n -1)
Find the eleventh term of the given geometric sequence.
3.) 5, -10, 20, -40,... ,
r = -
5
10
a1 = 5 a11= ?
n = 11
a11 = 5 • (-2) (11 -1)
a11 = 5 • (-2)10
a11 = 5 • 1024
a11 = 5 120
= - 2

Find the indicated term
1. a1 = 15, a2 = 3, a7=?
an = a1 • r (n -1)
a7 = 15 • (7 -1)
15
3
r = =
5
1
5
1
= 15 • (6)
5
1
= 15 •
15625
1
3125
3
a7 =
3125
3

Find the first term of the geometric sequence
1. a4 = -40, r = -2 n = 4
an = a1 • r (n -1)
-40 = a1 • (-2) (4 -1)
-40 = a1 • (-2)3
-40 = a1 • -8 Divide both sides by -8
a1 = 5

Find the first term of the geometric sequence
which the fourth term is -11 and which the
eleventh term is 11.
a4 = -11
-11= a1 • r (4-1)
-11 = a1 • r3
11 = a1 • r10
Substitute the fourth term and the 11th term in the formula
a11 = 11
n = 4 n = 11
Eq. 1 Eq. 2
11 = a1 • r(11 -1)
Divide Eq.2 by Eq. 1
11 = a1 • r10
-11 = a1 • r3
-1 = r7
r = -1
Substitute -1 for r in Eq. 1
-11 = a1 • (-1)3
-11 = a1 • (-1)
11 = a1

PRACTICE WORK
A.Find the eleventh term of each
geometric sequence.
1. 3. 5, 30, 180, 1080,….
2. ¼, ½, 1, 2,….
Find the indicated term
3. a3= 12, r = -4, a5=?
4. a3 = 12,a6 = 96, a11=?
5. a1 = -3, r = ?, a3 = -81

FIND THE SPECIFIED TERM OF EACH OF
THE FOLLOWING GEOMETRIC
SEQUENCES
 1. a1 =6 , r = 1/2, a9=?
 2. a1 = 6, a2 = 30, a8=?
 3. a1 = 36, a3 = 4, a11=?
 4. a3 = 12, a6 = 96 , a11=?
 5. How many terms are in the geometric
sequence 15,30, ….,240, 480?
sequence 6, 3,……, 3/64?

FIND THE SPECIFIED TERM OF EACH OF
THE FOLLOWING GEOMETRIC
SEQUENCES
 1. a1 =6 , r = 1/2, a9=?
 2. a1 = 6, a2 = 30, a8=?
 3. a1 = 36, a3 = 4, a11=?
 4. a3 = 12, a6 = 96 , a11=?
sequence 15,30, ….,240, 480?
sequence 6, 3,……, 3/128?

ACTIVITY: THE BIG DIFFERENCE
In the previous lesson, you learned what
arithmetic sequences are. In this activity, you will
now be asked to read an article entitled “Comparison
of arithmetic and Geometric Sequences.”
Please bear in mind the following reminders as you
read the article:
1. If there are words which are unfamiliar to you, look
for the meaning in the dictionary.
2. While you pay attention on the differences between
the two sequences, try to find some similarities as
well.

Now that you have finished reading the
article, let us find out if you are able to
complete the Venn Diagram below
completely.

Arithmetic
Sequence
Geometric
Sequence

 After completing the Venn diagram,
complete the following statements below:
 1. An arithmetic sequence and a
geometric sequence are similar because
they are both
________________________
 ________________________________.
 2.They are different because __________
 _________________________________
 _________________________________
_________________________________.

PROCESS QUESTIONS:
 1. What were presented in the article to help you
compare the two sequences?
 2. If you looked at only numbers, can you easily
detect the differences? Why?
 3. When you were asked to use formulas, were you
able to see the differences easily? Why? Why not?
 4. Did the graphs help you visualize the difference
between the two? How?
 5. Which among the three methods presented in
the article help you contrast the two sequences?
Why?

ACTIVITY: Can you tell?
 After finding the difference between the
arithmetic and geometric sequences, you
will now be asked to watch a video that
will help you fully understand how to
determine whether the sequence is
arithmetic or geometric.

PROCESS QUESTIONS
 How does one distinguish an arithmetic
sequence from a geometric sequence?
 How did the video help you fully
understand the difference between the
two?
 Can you find another way to make the
distinction apart from the one presented
in the video? Explain the process.

ASSIGNMENT
 As you go through the lesson, it is
important that you gain the correct
understanding of the terms that you will
be needing as the lesson progresses. Try
completing the table below by defining
some of the terms in the first column and
try to use each term in a meaningful
sentence.

TERMS OWN
DEFINITION
SENTENCE
1. sequence
2. Common
ratio
3.Recursive
formula
4. term
5. General
formula

ACTIVITY: 321 Chart
 After reading the two articles, let us see if you
can summarize your insights using the journal
below. In this activity, you will be asked to
complete the 321 chart regarding the you have
discovered.
321 Chart
Three things you found out:
1.
2.
3.
Two interesting things
1.
2.
One question I still have:
1.

ACTIVITY: The SEARCH is ON
 Now that you have learned what a geometric
sequence is, it is now time to find out how the
terms of a geometric sequence is derived.
 Answer the following process questions:
 1. How do you find the nth term of a geometric
sequence?
 2. How do you think the formula was arrived at?
 3. Is the formula presented applicable to the
examples presented/ how?
 4. How can outcomes of real life problems be
predicted?

Word Problems Involving
Geometric Sequence
1. We need to determine the number of
forwarded e-mails on the eighth round.
Five e-mails were sent on the first round.
Each of the five recipients sent five e-
mails on the second round, and so on.
2. A culture of bacteria doubles every 2
hours. If there are 500 bacteria at the
beginning, how many bacteria will there
be after 24 hours?

Geometric Sequence
 3. A mine worker discovers an ore sample containing
500 mg of radioactive material. It is discovered that
the radioactive material has a half life of 1 day. Find
the amount of radioactive material in the sample at
the beginning of the 7th
day?
 4. You complain that the hot tub in your hotel suite is
not hot enough. The hotel tells you that they will
increase the temperature by 10% each hour. If the
current temperature of the hot tub is 75°F, what will
be the temperature of the hot tub after 3 hours, to
the nearest tenth of a degree?

Geometric Sequence
 5. Ruby plans to bake the longest French
bread ever. She plans to cut them into six
pieces forming a geometric sequence,
where the shortest piece is 12 inches and
the longest is 384 inches. If she puts
together each end of the bread, how long
will the bread be?

ANSWER KEY
 5. Since we have the measures of the first and the last pieces of
bread, then we can have geometric sequence as follows:
 Pieces of bread:
 12 in, ___ , ____, ____, 384 in
 Having the first term, last term, and the number of terms in the
geometric sequence, we can find its common ratio by using the
general formula an =a1rn-1
 Given: a1 = 12, a6 = 384 and n = 6
 an = a1 r n-1
 384 = 12 (r 6 -1
) Replacement Property
 384 = 12 (r5
)
 32 = r5
MPE
 r = 2 Simplifying radicals
 Pieces of bread: 12 in, 24 in, 48 in, 96 in, 192 in, 384 in

Answer Key
 2. There are 500 bacteria to start, doubling every 2 hours
 Such bacterial growth is a geometric sequence with a common ratio of 2. The
number of hours, however is arithmetic with common difference of 2. Which term
number is 24? Find out by observation or:
 an = a1 + (n-1) d
 24 = 2 + (n-1)2
 n =12
 Now find the number of bacteria. The starting number and number of terms used
may vary
 an = a1 r n-1
 a12 = 500 (2) 12-1
= 1, 024, 000
500 1000 2000 ….. ?
Start After 2
hours
4 hours …… 24
hours
Term 1 2 …. ?

Answer Key
 3. 500 mg of ore. Half life of one day means that half of the
amount remains after 1 day
 Decide to either work with the “beginning” of each day, or the
“end” of each day, as each can yield the answer. Only the
starting value and number of terms will differ. We will use
“beginning”
 an = a1 r n-1
 a7 = 500 (½)7- 1
= 7.8125 mg
Begin of
day 1
500mg
Begin of
day 2
250 mg
Begin of
day 3
125 mg
……
End of day
1
250 mg
End of day
2
125 mg
End of day
3
62.5 mg
…..

Answer Key
 4. Starting temperature is 75°
 If the temperature is increased by 10%
the new temperature will be 110% of the
original temperature. The common ratio
will be 1.10. There are four terms
 75, after 1 hour, after 2 hours, after 3
hours
 an = a1 r n-1
 a4 = 75(1.10) 4-1
= 99.8°F

PROCESS QUESTIONS
 1. What were presented in the videos to help you
compare the two ways of solving for a geometric
series?
 2. How do the formulas reveal the difference
between the two?
 3. When will you know which type to use? How can
you tell?
 Complete the statement below:
 1. A finite geometric series and an infinite geometric
series are similar because
_______________________________.
 2. They are different because
_______________________________.

 1.a1 = 5 r = 5 a8=?
 a8= 5(5)8-1
= 5 (57
)
 = 390 625

GEOMETRIC SERIES
 It is now time to discover what a geometric
series is all about. To find out kindly read
pages 47-53 of your E-math book. Do it by
partner. Make sure to take note of the
significant difference between a geometric
sequence and series. Also, bear in mind
some formulas that are needed and the
purpose of these. Then answer the following
process questions on your graphing
notebook.

PROCESS QUESTIONS
 What is a geometric series?
 How is it different from a geometric
sequence?
 What are the two formulas that can be
used to solve for geometric series?
 What must be the reason why two
formulas are provided?

DEFINITION
 GEOMETRIC SERIES is the sum of a geometric sequence.
 FINITE GEOMETRIC SERIES are the sum of the first until
last term of the sequence.
 The sum of finite geometric series is calculated by the
formula
 Sn = t1 (1 – rn
)
 1 – r
 Where t1 = first term
 r = common ratio
 n = is the number of n

Examples
1.Find the sum of the first 12 terms of the
geometric sequence 3, -9, 27, -81,
243...a12.
2. Suppose Rico saves Php100 in January,
that each month thereafter he manages to
save one-half more than of what he saved
the previous month. How much is Rico’s
savings after 10 months?

 INFINITE GEOMETRIC SERIES is an
infinite series (has no last term
given)whose successive terms have a
common ratio. This series converges if
and only if the absolute value of the
common ratio is less than 1 |r| < 1
 S ∝ = t
 ___________
 1 – r

EXAMPLES
 1. Find the sum of the geometric
sequence
 a. ½, ¼, 1/8, 1/16,.....
 b. 3, 1, 1/3, 1/9,....

EXAMPLES
 Directions: Insert the indicated number of
geometric means between the given
numbers. Write the resulting geometric
sequence.
 a. three between 2 and 512
 b. two between -5 and 1080
 c. three between 7 and 567

 1. Jane at 8 a.m. got a message that all 1093 school pupils
would go to the cinema. Within 20 minute she said it to
the three friends. Each of them again for 20 minutes said
to the other three. In this way, the message spread
further. At what time do all the schoolchildren know they
will go to the cinema?
 a1 = 1 r = 3 an = 1 093
 1 093 = 1 (3) n - 1
 log 1 093 = n - 1 log 3
 n - 1 = log 1 093/log 3
 n - 1 = 6.37
 n = 7 7 x 20 = 1 40/ 60 = 2 hours and 20
minutes . At 10:20 a.m. the school children will go to the
cinema.


EXERCISES: Solve the given
problem.
 1. Jane at 8 a.m. got a message that all 1093 school pupils
would go to the cinema. Within 20 minute she said it to
the three friends. Each of them again for 20 minutes said
to the other three. In this way, the message spread
further. At what time do all the schoolchildren know they
will go to the cinema?
 2. Ms. Cruz has signed a 5-year professional consultant
contract with a beginning salary of P2 000 000.00 per year.
The management gives her the following options with
regard to her salary for the next 5 years.Which option will
give more income?
 A. An annual increase of Php 100 000.00 per begining after
1 year.
 B. An annual increase of 5% per year beginning after 1
year.
 C. A bonus of Php 105 000.00 each year?

EXERCISES: Solve the given
problem.
 1. Find the sum of the given geometric
sequence -6, -2, -2/3,.....
 2. A pendulum swings through an arc of length
100 cm. If each subsequent swing is 90% of the
preceding swing, what is the total distance it
travelled by the time it stops completely.
 3. A golf bounces 80% of the height from which
it falls if it is dropped onto a marble surface
from a height of 100 ft. Find the total distance it
has travelled by the time it comes to rest.

GRADED ACTIVITY
Solve the given geometric sequence or
series problems.Show your complete
solutions.
1. At the end of each year, the value of a car depreciates
by of its original value. What is its value at the end of its
sixth year if it originally costs Php1.2 million?
2. An antique jewelry worth Php25, 000 at the beginning
of 2005 increases in value by 10% every year. At the end
of what year will its value be
Php40 262.75?
3. The number of bacteria in a certain culture triples
every 6 h. If initially there were 65000 bacteria, how
many bacteria will there be after 72 h?

GRADED ACTIVITY
Solve the given geometric sequence or series
problems.
4. An insurance salesman makes Php15,000
commission on sales in his first year? Each
year, he increases his sales by 50%. How
much commission would he make in total over
6 years?
5. A rubber ball is dropped from a height of 60
m on each rebound it rises two thirds of the
height from which it last fell. Find the distance
travelled by the ball before it comes to rest.

Geometric Series and Sequences || Grade 10 First Quarter

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