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Mathematics
00 11

Quarter 1 - Module 3

Determining Arithmetic Means and

nth Term of an Arithmetic Sequence

Department of Education ● Republic of the Philippines

Sub-topic: Arithmetic Mean

Finding a certain number of terms between two given terms of an arithmetic sequence is a
common task in studying arithmetic sequences. The terms between any two nonconsecutive terms of
an arithmetic sequence are known as arithmetic means.
Arithmetic means are the terms between any two nonconsecutive terms of an arithmetic
sequence.
It is necessary to solve the common difference of an arithmetic sequence to insert terms
between two nonconsecutive terms of an arithmetic sequence. The formula for the general term of an
(x+y)
arithmetic sequence, an = a1 + (n – 1)d and the mid-point between two numbers, can also be
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used.

Example 1: Insert 4 arithmetic means between 5 and 25.

Solution: Since we are required to insert 4 terms, then there will be 6 terms in all.
Let a1 = 5 and a6 = 25. We will insert a2, a3, a4, a5 as shown below:
5, a2, a3, a4, a5, 25

We need to get the common difference. Let us use a6 = a1 + 5d to solve for d. Substituting the
given values for a6 and a1, we obtain 25 = 5 + 5d.
25 = 5 + 5d
25 – 5 = 5d
20 = 5d
4 = d
So, d = 4.
Using the value of d, we can now get the values of a2, a3, a4, and a5. Thus, a2 = 5 + 4(1) = 9,
a3 = 5 + 4(2) = 13, a4 = 5 + 4(3) = 17, and a5 = 5 + 4(4) = 21.
The 4 arithmetic means between 5 and 25 are 9, 13, 17, and 21.
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Example 2: Insert three terms between 12 and 56.

Solution: Since we are required to insert 3 terms, then there will be 5 terms in all.
Let a1 = 12 and a5 = 56. We will insert a2, a3, a4 as shown below:
12, a2, a3, a4, 56

We need to get the common difference. Let us use a5 = a1 + 4d to solve for d. Substituting the
given values for a5 and a1, we obtain 56 = 12 + 4d.
56 = 12 + 4d
56 – 12 = 4d
44 = 4d
11 = d
 Therefore, the common difference is 11.
Using the value of d, we can now get the values of a2, a3, and a4. Thus, a2 = 12 + 11(1) = 23,
a3 = 12 + 11(2) = 34, and a4 = 12 + 11(3) = 45.
The 3 arithmetic means between 12 and 56 are 23, 34, and 45.

ACTIVITY 1
A. Answer the following.
1. Insert two arithmetic means between 20 and 38.
2. Insert three arithmetic means between 52 and 40.
3. Find the missing terms of the arithmetic sequence 5, ___, ___, ___, ___, 25.
4. Find the missing terms of the arithmetic sequence 0, ___, ___, ___, ___, ___, 15.
5. The fifteenth term of an arithmetic sequence is –3 and the first term is 25.
Find the common difference and the tenth term.

B. Use the following numbers inside the box to complete the arithmetic sequence below.
You may use a number more than once.

6 7 10 11 12
13 15 18 19 21
11/2 17/2 23/2 27/2 29/2
1. 2, ___, ___, 14
2. 4, ___, ___, ___, 10
3. 6, ___, ___, ___, 16
4. 9, ___, ___, ___, ___, 24
5. ___, 17, ___, ___, 11

C. Answer the following problems.

1. Flower farms in Monte Alegre grew different variety
of flowers including anthurium. Monica, a flower
arranger, went to Monte Alegre to buy anthurium.
She plans to arrange the flowers following an
arithmetic sequence with four (4) layers. If she put
one (1) anthurium on the first layer and seven (7)
on the fourth layer, how many anthurium should be
placed on the second and third layer of the flower
arrangement?
2. In some of the kiddie parties nowadays, tower cupcakes
were quite popular because it is appealing and less
expensive. In Juan Miguel’s 1st birthday party, his mother
ordered a six (6) layer tower cupcakes. If the 1st and
4th layer of the tower contains 6 and 21 cupcakes,
respectively, how many cupcakes are there in the 6 th layer
(bottom) of the tower assuming arithmetic sequence in the
number of cupcakes.
Sub-topic: Sum of Arithmetic Sequence

The sum of terms in an arithmetic sequence can be solved using these formula:
n
Sn = ( a + a n ), given the 1st and last term of the sequence or
2 1
n
Sn =
2
[ 2 a 1 + ( n - 1 ) d ], given the first term and the common difference.

THE SECRET OF KARL

What is 1 + 2 + 3 + ... + 50 + 51 + ... + 98 + 99 + 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? Can you beat that? His name was Karl Friedrich Gauss (1777-1885). Do you know
how he did it? Let us find out by doing the activity below.

Determine the answer to the above problem. Then look for a partner and compare your answer
with his/her answer. Discuss with him/her your technique (if any) in getting the answer quickly. Then
with your partner, answer the questions below and see if this is similar to your technique.
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 answers 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?
Solutions:
The first term or a1 is 1 and the last term or an is 100. 100 is the 100th term of the sequence,
therefore n is 100. The common difference is 1.
Solving for the sum of all the terms of the arithmetic sequence, we have
n n
Sn = 2 (a1 + an ) or Sn = 2 [ 2 a1 + ( n - 1) d ]

100 100
S100 = ( 1 + 100 ) S100 = [ 2(1) + ( 100 - 1 ) 1 ]
2 2

S100 = 50 ( 101 ) S100 = 50 [ 2 + 99 ] = 50 [101]

S100 = 5050 S100 = 5050

The sum of the integers from 1 to 100 is 5050.

More Examples
Example 1: Find the sum of the first 10 terms of the arithmetic sequence 5, 9, 13, 17, ...
n
Solution: Sn = 2 [ 2 a 1 + ( n - 1) d ]

10
S10 = [ 2(5) + ( 10 - 1 ) 4 ]
2
S10 = 5 [ 10 + (9)4 ] = 5 [ 10 + 36 ]
S10 = 5 [46] = 230

Example 2: Find the sum of the first 20 terms of the arithmetic sequence -2, -5, -8, -11, ...
n
Solution: Sn = 2 [ 2 a 1 + ( n - 1) d ]

20
S20 = [ 2(-2) + ( 20 - 1 ) -3 ]
2
S20 = 10 [ -4 + (19)-3 ] = 10 [ -4 - 57 ]
S20 = 10 [-61] = -610

ACTIVITY 2
A. Find the sum of each of the following.
1. integers from 1 to 50
2. odd integers from 1 to 100
3. first 25 terms of the arithmetic sequence 4, 9, 14, 19, 24, …
4. first 20 terms of the arithmetic sequence –16, –20, –24, ...
5. first 10 terms of the arithmetic sequence 10.2, 12.7, 15.2, 17.7, ...

B. Find Sn for each of the following given.

1. 6, 11, 16, 21, 26, 31, 36, 41, 46; S9
2. 10, 15, 20, 25, ...; S20
3. a1 = 25, d = 4; S12
4. a1 = 65, a10 = 101; S10
5. a4 = 41, a12 = 105; S8

C. Answer the following problems.

1. A store sells Php1000 worth of suman during its first
week. The owner of the store has set a goal of
increasing her weekly sales by Php300 each week.
If we assume that the goal is met, find the total sales
of the store during the first 15 week of operation.
2. Find the seating capacity of a movie house with
40 rows of seats if there are 15 seats on the first row,
 18 seats in the second row, 21 seats in the third row
and so on.

Name and Signature of Student: _______________________________ Section: ______________