INTERACTIVE MULTIMEDIA on ARITHMETIC SEQUENCE_SIM FOR MATH GRADE 10.pdf
3. Welcome to Module 3!
This module is divided in to three lessons:
o Lesson 1 – nth term of an Arithmetic Sequence
o Lesson 2 – Arithmetic Means
o Lesson 3 – Sum of the Terms of a Given Arithmetic
sequence of a given Arithmetic
Sequence
Reminder: Use this module with care.
Next
4. Next
After going through this
module, you are expected
to:
1. determine the nth
term of an arithmetic
sequence;
2. define arithmetic mean;
3. find the sum of the terms
of a given arithmetic
sequence.
7. In Lessons 1 & 2, you have learned to illustrate sequences and
give the nth term or the rule for a particular sequence. These are
set of objects or numbers arranged in sequence order.
Example:
Write the first 5 terms of a sequence given the nth term
an = 3n – 1.
Let a1, a2, a3, a4, and a5 be the first 5 terms. Thus,
n = 1, 2, 3, 4, and 5.
a1 = 3(1) - 1 = 2
a2 = 3(2) – 1 = 5
a3 = 3(3) – 1 = 8
a4 = 3(4) – 1 = 11
a5 = 3(5) – 1 = 14.
The first 5 terms of the nth term an = 3n – 1 are 2, 5, 8, 11,and 14.
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8. This lesson focuses on the nth term of an arithmetic sequence, arithmetic
means,
and the sum of the terms of a given arithmetic sequence. Read the
problem
below and answer the questions that follow.
Try this:
Determine if the following series of numbers are arithmetic sequences or
not. If the given is an arithmetic sequence, write A on the space provided.
If not, write N.
______ 1. 2, 4, 6, 8, 10, 12, 14
______ 2. 5, 4, 7, 9, 11, 10, 8
______ 3. 25, 28, 31, 34, 37, 40, 43
______ 4. 14, 15, 17, 17, 19, 20, 21
______ 5. 124, 126, 128, 130, 132, 134, 136
Compare your answers with those found in the Answer Key.
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9. Determine if the following series of numbers are arithmetic
sequences or not. If the given is an arithmetic sequence,
write A on the space provided. If not, write N.
A 1. 2, 4, 6, 8, 10, 12, 14
N 2. 5, 4, 7, 9, 11, 10, 8
A 3. 25, 28, 31, 34, 37, 40, 43
N 4. 14, 15, 17, 17, 19, 20, 21
A 5. 124, 126, 128, 130, 132, 134, 136
If you got 3 to 5 correct answers, you’re doing great. You
can proceed to the next lesson. If you got only 2 or less, read
Module 2 again and after doing that, try to solve the
exercises given above again.
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10. EXAMPLE 1
Consider the sequence 2, 6, 10, 14, . . . . What is the 12th term in the given sequence?
Follow the steps below to solve this problem.
STEP 1 Find the common difference.
6 – 2 = ____
10 – 6 = ____
14 – 10 = ____
The common difference is: d = 4.
STEP 2 Determine the 1st term in the given arithmetic sequence.
The 1st term in the given arithmetic sequence is 2. This means that a1= 2.
STEP 3 Find the symbol for the unknown term in the sequence.
You are asked for the 12th term in the given arithmetic sequence. Thus, the symbol for the
unknown term is a12.
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11. STEP 4 Write the equation or the number sentence for the unknown term in the sequence.
The equation for a12 is:
a12 = a1+ (12 – 1) d
a12 = a1+ 11d
STEP 5 Substitute the values in the equation and solve for the answer.
From Step 1, d = 4 and from Step 2, a1 = 2. Thus we have
a12 = a1+ 11d
a12 = 2 + 11 (4) = 46
This means that the 12th term of the arithmetic sequence 2, 6, 10, 14, . . . . . .is 46.
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12. REMEMBER:
To solve for the nth term in an arithmetic sequence, we use the formula:
an = a1 + (n–1)d where an = nth term
a1 = 1st term
n = number of terms
d = common difference
♦ We follow the following steps in using the equation above:
STEP 1 Find the common difference.
STEP 2 Determine the 1st term in the given arithmetic sequence.
STEP 3 Find the symbol for the unknown term in the sequence.
STEP 4 Write the equation for the unknown term in the sequence.
STEP 5 Substitute the values in the equation and solve for the result.
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13. REMEMBER:
One of the common tasks in studying arithmetic sequence is finding the arithmetic means. The
arithmetic means are the terms between any two non-consecutive terms of an arithmetic
sequence.
Examples: a.) Insert 2 arithmetic means between 4 and 19.
Solution: Since we are required to insert 2 terms, then there will be 4 terms in all.
Step 1: Let a1 = 4 and a4 = 19. Then we will insert a2, a3 as shown below:4, a2, a3, 19
Step 2: Get the common difference. Let us use a4 = a1 + 3d
Step 3: Substitute the given values of a4 and a1. That is 19 = 4 + 3d.
Step 4: Solve for d.
19 – 4 = 3d
15 = 3d
5 = d
Step 5: Using the value of d, we can now get the values of a2, and a3. Thus,
a2 = 4 + 5(1) = 9
a3 = 4 + 5(2) = 14
Answer: The 2 arithmetic means between 4 and 19 are 9 and 14.
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14. Shown in the table is a list of the number of bottles of juice drinks
Jaine-Anne sold from Monday to Saturday.
PROBLEM: How many bottles of soft drinks will Jaine-Anne
sell in 10 days?
From the series of numbers, find out the following:
1. Is the series an arithmetic sequence?
2. Do the numbers in the series have a common difference? If yes, what is the common difference or d?
3. What is the first term or a1?
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15. From the series of numbers, find out the following:
1. Is the series an arithmetic sequence?
2. Do the numbers in the series have a common difference? If yes,
what is the common difference or d?
3. What is the first term or a1?
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Day No. of Bottles
1 20
2 70
3 120
4 170
5 220
6 270
16. The symbol for sum is S. then for the sum of:
✓ two terms is S2
✓ three terms is S3
✓ four terms is S4
✓ five terms is S5
✓ six terms is S6 ,.. What is the symbol for the sum of 7 terms? 10 terms? 12 terms?
If your answer are S7, S10 and S12 , you are correct!
Here’s the formula to find the sum of the terms of an arithmetic sequence:
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1
17. Given:
a1 = 20
n = 10
Solve for S6.
Formula:
Solution:
S10 = [2(20) + (10-1)50]
=5 [40 + (9)50]
=5 [40 + 450)
=5 (490)
=2,450 Next
10
2
Answer: There are 2,450 bottles of soft drinks will
Jaine-Anne sell in 10 days.
1
66. 1. What are the steps in finding the nth
term of an arithmetic sequence?
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2. What is an arithmetic mean?
3. What is the formula to find the
sum of the terms in an
arithmetic sequence?