​Solution:

​

​a₁ = Php 60, 000 d = Php 3,600 n = 5 years

​S_n = n [2a₁ + (n-1)d ] S₅ = 5 [120, 000 + 14,400 ]

​ 2 2

​S₅ = 5 [2(60,000) + (5-1)3,600 ] S₅ = 5 [134,400 ]

​ 2 2

​S₅ = 5 [120, 000 + (4)3,600 ] S₅ = 5 [67, 200 ]

​ 2 S₅ = 336, 000

​​S₅ = 5 [120, 000 + 14,400 ]

​ 2

MODULE 5: GEOMETRIC SEQUENCES VS. ARITHMETIC SEQUENCES​

​LEARNING COMPETENCY:

​The learner illustrates Geometric Sequence M10AL-Id-1 and the learner differentiates a geometric sequence from an arithmetic sequence. M10AL-Id-2

​LEARNING OBJECTIVES:

​After going through this module, the learner should be able to:

a. determine a geometric sequence;

b. identify the common ratio of a geometric sequence;

c. find the missing term of a geometric sequence;

d. determine whether a sequence is geometric or arithmetic; and

e. explain inductively the difference between arithmetic sequence and geometric sequence using real life-situations.

​WHAT I KNOW

​Find out how much you already know about the topic. Answer the Quizizz link sent by your teacher for live game, otherwise, use this link https://quizizz.com/join?gc=64699422

​WHAT I KNOW

Multiple Choice. Read and analyze the following items and determine the letter of the correct answer from the given choices. Write the CAPITAL letter of the correct answer on a separate sheet of paper.

1. What operation is used in obtaining the next terms in a geometric sequence?

A. Addition C. Division​

B. Multiplication D. Subtraction​

​WHAT I KNOW

​2. What operation is used in obtaining the succeeding terms in an arithmetic sequence?

A. Addition C. Division

B. Multiplication D. Subtraction

​WHAT I KNOW

​3. Which of the following terms is closely related to a geometric sequence?

A. Common Ratio C. Both A & B

B. Common Difference D. None of these.

​WHAT I KNOW

​4. Which of the following terms is closely associated to an arithmetic sequence?

A. Common Ratio C. Both A & B

B. Common Difference D. None of these.

​WHAT I KNOW

​5. Which of the following sequences is arithmetic?

A. 12, −12, −12, −12, … C. 12, 25/2 , 13, 27/2 , …

B. 32, 8, 2, 12 , … D. 1, 2, 3, 5, 8, …

​WHAT I KNOW

​6. Which of the following presents a geometric sequence?

A. 12, −12, −12, −12, … C. 12, 25/2 , 13, 27/2 , …

B. 32, 8, 2, 12 , … D. 1, 2, 3, 5, 8, …

​WHAT I KNOW

​7. What is the common difference in the sequence −1, − 1/2, 0, 1/2 , 1, …?

A. −1 C. 0

B. − 1/2 D. 1/2

​WHAT I KNOW

​8. What is the common ratio in the sequence −1, 1/2, − 1/4 , 1/8 , …?

A. −1 B. − 1/2 C. 0 D. 1/2

​

​WHAT I KNOW

​For items 9 to 15, determine whether the given situation presents an arithmetic sequence, a geometric sequence, or neither. Write A for arithmetic, G for geometric, and N for neither.

​

​WHAT I KNOW

​9. An amount of money invested in a bank that offers an interest rate of 6% compounded annually

​

​WHAT I KNOW

​10. Saving ₱5 in your piggy bank daily

​

​WHAT I KNOW

​11. Pushing a swing with a constant force

​

​WHAT I KNOW

​12. 1, 2, −1, 3, −2, …

​WHAT I KNOW

​13. Half-life of Carbon

​WHAT I KNOW

​14. Graph of a linear function

​WHAT I KNOW

​15. A car running with a constant speed

​LESSON 1:

GEOMETRIC SEQUENCES

​Consider yourself depositing an amount of money in MMR Bank which offers 8% interest rate compounded annually. If you deposited Php 10,000 in that bank, how much will you earn after a year? 2 years? 3 years?

​Let us compute for the simple interest which is only paid on the initial principal amount.

​(principal amount) + (interest earned) = (end of the year balance)​Php 10, 000 + [(Php 10,000)(0.08)] = Php 10, 800.00

​How much will you earn in two years?

​Php 10, 000 (1.08)(1.08) = Php 11, 664.00

​How much will you earn in three years?

​Php 10, 000 (1.08)(1.08)(1.08) = Php 12, 597.12

​GEOMETRIC SEQUENCE

​a sequence obtained by multiplying a common ratio to the preceding terms in order to obtain the succeeding terms.

​Common Ratio ( r ) 

​WHAT's more

​Find the next three terms of the geometric sequence 3, 21, 147,...

Solution:

a. First, identify the common ratio (r)

Common Ratio (r) is obtained by

​

Solution:

b. Afterwards, multiply the obtained common ratio (r) to the preceding term to get the next term

​

​Therefore, the next three terms of the geometric sequence are 1029, 7203, and 50, 421

​WHAT's more

​You are planning to buy a new blouse which costs Php 300 as a present to your mother this Christmas season. You started saving money on the first week of November and doubled the amount to be saved every week. If you started saving Php 5 on the first week, will you be able to buy the blouse at the end of the second week of December? What is the sequence obtained?

​The geometric sequence obtained is 5, 10, 20, 40, 80, 160,...

​If you do this scheme of saving from the first week of the year until the very last week with Php 1 as the initial amount to be saved, how much will you be able to save for a year? What will you do with such earning?

Your Turn!!!

Solution:

Solution:

​The missing term is a succeeding term and comes after 48

​Therefore, the missing term in the geometric sequence is 192.

Solution:

​The missing term is a preceding term and comes before 32.

​Therefore, the missing term in the geometric sequence is 16.

Your Turn!!!

Your Turn!!!

Your Turn!!!

Your Turn!!!

Your Turn!!!

Your Turn!!!

Your Turn!!!

Your Turn!!!

Your Turn!!!

Your Turn!!!

what is it?

what is it?

remember:

​An Arithmetic Sequence is a sequence obtained by adding a common difference (d) to a preceding term in order to obtain the next term.

what is it?

what's more?

​ARITHMETIC OR GEOMETRIC

​Determine whether the sequence is Geometric or Arithmetic

Solution:

​First, let's investigate if a common ratio or a common difference exists.

​Therefore, we can conclude that the sequence is an arithmetic sequence.

​Let's try to obtain a common difference by using the first three terms 12, 15 and 18.

Solution:

Solution:

Solution:

​WHAT I CAN DO

​ INVESTIGATE AND CONCLUDE

Read and analyze the situation below. (Write your answer on a clean sheet of paper. NO, SOLUTION, NO POINT.

Consider yourself depositing an amount of money in a local bank which offers 5% interest rate compounded annually. If you deposited ₱10,000 in that bank, how much will you earn after 3 years?

1. What kind of sequence is illustrated in the above situation?

2. If it shows an arithmetic sequence, then what is the common difference? On the other hand, if it shows a geometric sequence, then what is the common ratio? Provide a justification for your answer using any method.

​ADDITIONAL ACTIVITY

​Ben is planning to visit his friend Albert who is suffering from colds and headache. Ben uses his car to travel to Albert’s house. The initial speed of Ben’s car is 20 mph, then constantly increasing one-fourth of its preceding speed.

1. If the constant increase of its speed is done every 10 minutes, what will be the speed of his car for the first 30 minutes?​

2. Construct a table of values which shows the relationship between the speed of the car for the first hour if the constant increase is 13 of the preceding speed for every 20 minutes.

3. What is the common ratio obtained by the sequence based from the table of values?

MODULE 6: FINDING THE nth TERM OF A GEOMETRIC SEQUENCE AND GEOMETRIC MEANS​

​LEARNING OBJECTIVES:

After going through this module, the learner should be able to:

a. familiarize the formulas in finding terms of geometric sequence,

b. find the 𝑛th term of a geometric sequence, and

c. determine the geometric mean/s of a geometric sequence.​

​Determine how much you already know about Geometric Sequences and Geometric means. Answer the Quizizz link sent by your teacher for live game, otherwise, use this link https://quizizz.com/join?gc=12526110

​

2. What is the common ratio of the sequence 36, 12, 4, …?

A. 3 C. 1/3

B. – 3 D. – 1/3​

​3. The first term of a geometric sequence is 5, and the second term is 10, what is the sixth term?

A. 160 C. 320

B. 80 D. 640​

​4. The common ratio of a geometric sequence is 1/2 , and the fourth term is 1/4 . What is the third term of the sequence?

A. 1 C. 1/4

B. 1/2 D. 1/8​

​For items 6 – 10, find the specified term of the following geometric sequences.​

​6.) 3, 6, 12, … find a7​

​7.) 10, 5, 5/2, … find a6​

​8.) 1, 3, 9, … find a9​

​9.) 2, – 1 , 12, … find a10​

​10.) 100, 20, 4, … find a8​

​For items 11 – 15, insert geometric means in each geometric sequence.​

​11.) 3, ____, 8

​13.) 7, _____, _____, 56

​14.) 36, ____, ____, ____, 9/64

​15.) – 1, _____, – 9 , _____, – 81

​LESSON 3:

FINDING THE nth TERM OF A GEOMETRIC SEQUENCE

Look at the sequences below. Find the next three terms of the geometric sequences

​5, 10, 20,...

​36, 12, 4,...

The sequence 5, 10, 20, ... is a geometric with common ratio of 2. Therefore, the next three terms are 40, 80, and 160

What is the 7th term of the sequence 5, 10, 20,...? 

Solution:

​Find the specified term of the given geometric sequence.

12 , 2, 8,…; find 𝑎6

​The given values will then be substituted to the formula in finding the 𝑛th term of a geometric sequence.

​

Solution:

​Find the specified term of each of the given geometric sequences. Show your complete solutions.

1.) 3,6,12,… 𝑎7​

​

​Find the specified term of each of the given geometric sequences. Show your complete solutions.

2.) 18,6,2,… 𝑎9​

​

​Find the specified term of each of the given geometric sequences. Show your complete solutions.

2.) 18,6,2,… 𝑎9​

​

​Find the specified term of each of the given geometric sequences. Show your complete solutions.

3.) 1,12 ,14 ,… 𝑎7

​