Math10 Module Q1 Wk5
Math10 Module Q1 Wk5
Math10 Module Q1 Wk5
Mathematics
Quarter 1 Module: Week 5
Problems Involving Sequences
1
What I Need to Know
The module provides you activities that will help you learn about solving problems
involving sequences.
After going through this module, you are expected to:
1. Identify and describe the pattern of each sequence problem
2. Solve systematically the sequence problem
3. Relate problem in real life situation.
What I Know
Pre - Test
Choose the letter of the best answer. Write the chosen letter on a separate sheet of
paper.
1. Describe the pattern in the sequence 13, 15, 17, 19, ..., and find the next three terms
A. Add 2; 23, 25, 27. C. Add –2; 17, 15, 13.
B. Multiply by 2; 38, 76, 152. D. Add 2; 21, 23, 25.
2. Describe the pattern in the sequence 4, 8, 16, 32, ..., and find the next three terms
A. Multiply by 2; 64, 128, 256. C. Multiply by 2; 128, 256, 512.
B. Multiply by –2; –64, 128, –256. D. Add 2; 34, 36, 38.
3. The table shows the predicted growth of a bacteria after various numbers of hours. What is
the explicit formula for the sequence of the number of bacteria?
Hours (n) 1 2 3 4 5
Number of
19 38 57 76 95
Bacteria
1 1
10. What is the sixth term of a harmonic sequence if a1 is and d is ?
3 5
2
1 1 1 1
A. B. C. D.
8 13 23 28
Lesson
Problems Involving Sequences
1
What’s In
Choose the correct answer from the box. Write your answers in separate sheet
of paper.
6 29 31 32 91 165
th
1. Find the 14 term of the arithmetic sequence 5, 7, 9, 11, …
2. Find the sum of the first ten terms of the arithmetic sequence 5, 6, 9, …
3. Find the 5th term of the geometric sequence 2, 4, 8, …
4. Find the geometric mean between 3 and 12
5. Find the arithmetic mean between 9 and 49.
What’s New
Try Me!
What is 1 + 2 + 3 + …+ 50 + 51 + …+ 98 + 100?
A famous story tells that this was the problem given by an elementary school teacher to
a famous mathematician to keep him busy. Do you know that he was able to get the sum
within seconds only? His name was Karl Friedrich Gauss (1777-1885). Do you know that he
did it? Find out by doing the activity below.
Determine the answer to the above problem.
1 + 2 + 3 + …+ 50 + 51 + …+ 98 + 99 + 100
Questions:
1. What is the sum of each of the pairs 1 and 100, 2 and 99, 3 and 98, … 50 and 51?
2. How many pairs are there in # 1?
3. From your answer in # 1 and # 2, how do you get the sum of the integers from 1 to
100?
4. What is the sum of the integers from 1 to 100?
3
What is It
The following guide will help you to solve words problems involving
sequence:
1. Read and carefully analyze the problem.
2. Identify the corresponding formula in solving the problem.
3. Solve carefully and systematically what is asked on the problem.
Example 1
Cherry made deposits from her school allowances as follows: Php 10 on the first week,
Php 13 on the second week, Php 16 on the third week, and so on, until she made 52 deposits.
What was amount of her last deposit? How much was her total savings after 52 deposits?
Solution:
Step 1. 10, 13, 16, … Form an arithmetic sequence
Step 2. a1 = 10;
d = 13 – 10 or 16 – 13
Given
d = 3;
n = 52
Step 3. a52 = ___; S52 = ___ Identify what is asked
Step 4. 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 Corresponding Formula to find the last term
Step 5. 𝑎52 = 10 + (52 − 1)3 Substitution
Step 6. 𝑎52 = 10 + (51)3 Simplification
Step 7. 𝑎52 = 10 + 153
Step 8. 𝒂𝟓𝟐 = 𝟏𝟔𝟑 52nd term
Step 9. n
Sn =
2
[ 2 a1+ ( n−1 ) d ] Corresponding Formula to find the sum of all terms
Step 10. 52
S52 = [ 2(10)+ ( 51−1 ) 3 ] Substitution
2
Step 11. 𝑆52 = 26[20 + (51)3] Simplification
Step 12. 𝑆52 = 26[20 + 153]
Step 13. 𝑆52 = 26[173]
Step 14. 𝑺𝟓𝟐 = 𝟒𝟒𝟗𝟖 Sum of first 52 terms
Remember!
In solving word problems involving arithmetic sequence, the following
formula can be used:
Example 2
The first term of a geometric series is 3, the common ratio r is 2, the nth term is 48,
and the sum of the n terms is 93. Find the number of terms n.
Solution:
Step 1. a1 = 3; Given
an = 48;
r=2
Step 2. n = ___ Identify what is asked
Step 3. 𝑎𝑛 = 𝑎1𝑟𝑛−1 Corresponding Formula to find the number of terms
Step 4. 48= 3(2)𝑛−1 Substitution
Step 5. 48 3(2)n−1
3
= Eliminate 3 by MPE
3
Step 6. 16= (2)𝑛−1 Simplification
4
Step 7. 24= (2)𝑛−1 Rewrite 16 to exponential form 24 (with the base of 2) so
that it is the same to 2𝑛−1
Step 8. 4=n-1 Since both sides have the equal bases (2 = 2)
Step 9. 4+1=n Simplification
Step 10. n=5 There are 5 terms.
Remember!
In solving word problems involving geometric sequence, the following
formula can be used:
a1
Infinite S=
1−r
Example 3
Antonio drives 10 km from Silang to Dasmariñas at 40 km per hour and returns at
the rate of 60 km per hour. What is the average rate for the entire round trip?
Solution:
Step 1. Rate 1 = 40 km per hour Given
Rate 2 = 60 km per hour
Step 2. What is the average rate Identify what is asked
for the entire round trip?
Step 3. 2ab Corresponding Formula (Average of two rates)
H=
a+b
Step 4. 2(40)(60) Substitution
H=
40+ 60
Step 5. 4800
H= Simplification
100
Step 6. H = 48 km/hr The average of two rates
Remember!
Harmonic mean is related to finding average. It is generally used in finding
the average of ratios, rates of increase, index number, etc.
2ab
If the average of two rates is asked, use H=
a+b
3 abc
If the average of three rates is required, use H=
ab+ ac+bc
What’s More
Solve each problem carefully and systematically. Write your answers with your
solution on a separate answer sheet.
1. Ferdinand saved 10 pesos on the first day of the year, 12 pesos on the second day, 14
pesos on the third day, and so on, up to the end of the year. How much did he save on the
365th day?
Solution:
Step 1: 1st day ___, 2nd day ___, 3rd day ___, … Given
Step 2: d = ___ common difference
Step 3: __________________ What is the corresponding formula?
5
Step 4: __________________ Use substitution method
Step 5: __________________ Simplify (Use GEMDAS)
Step 6: 365th term = _____ What is asked on the problem
Use the problem to complete the table below for the student’s savings of a week.
A student saved Php 10 on Sunday and doubled his savings each day thereafter.
How much did he save on the 7th day? What was his total savings for the week?
What I Can Do
Problem Solving
Solve the following problem with logical solution. Write your answers in a separate sheet of
paper.
Mr. Castro gave his daughter Php 300 on her 10th birthday and intends to increase
this by Php 200 each year. How much will the daughter receive on her 18th birthday?
Assessment
Read and carefully analyze the problem below. Choose the letter of the correct answer. Write
your answers in a separate sheet of paper.
Aida gets a starting salary of Php 6,000 a month, and an increase of Php 600
annually. What will be her salary during the fifth year?
6
a. 11400 b. 10800 c. 10200 d. 9600
Additional Activities
What new realizations do you have about the topic? What new connections have
you made for yourself? What questions do you still have? Copy and fill-in the
Learned, Affirmed, and Challenged cards given below.
Answer Key
What’s More
What’s In
What I Know 1.
1. 31 Step 1. 10, 12, 14
1. D 6. C 2. 165 Step 2. 2
2. A 7. C 3. 32 Step 3. an = a1 + (n – 1) d
3. A 8. A 4. 6 Step 4. a365 = 10 + (365 – 1) 2
4. A 9. D 5. 29 Step 5. a365 = 10 + 728
5. C 10. D Step 6. a365 = 738
2.
Step 1. 8, 16, 32
What I Have Learned Step 2. 2
Step 3. an = a1r n-1
Monday 20 Step 4. 32 = 8(2) n-1
Tuesday 40 Step 5. 22 = 2n-1
Wednesday 80 Step 6. 3
Thursday 160
Friday 320
Saturday 640 Assessment
Sunday 1280
Total 2540 1. a
1. 2. b
1. 20 3. b
2. 1280 4. d
3. 2540 5. b
7
References:
Next Century Mathematics 10 p. 42-64
Learners Module in Mathematics 10 p. 26-44
Intermediate Algebra II p. 172-200