The document discusses arithmetic sequences, including definitions and formulas for calculating terms, common differences, and sums. It provides examples and problems to determine specific terms, common differences, and arithmetic means within sequences. Additionally, it outlines methods for finding sums of arithmetic sequences and includes practice problems for the reader.

1. Arithmetic Sequence
Arithmetic Sum
Arithmetic Mean
Prepared by:
Maricel T. Mas
T-I/ Lipay high School

2. Formula:
an = a1 +(n-1)d
Where:
𝑎1= 𝑓𝑜𝑟 𝑡ℎ𝑒𝑓𝑖𝑟𝑠𝑡𝑡𝑒𝑟𝑚
𝑎𝑛 = 𝑓𝑜𝑟 𝑡ℎ𝑒𝑛𝑡ℎ 𝑡𝑒𝑟𝑚
𝑑 = 𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
𝑛 =
𝑓𝑜𝑟 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑟𝑚 𝑓𝑟𝑜𝑚 𝑎1𝑡𝑜 𝑎𝑛

3. Sample Problem
1. Find the 5th term and 11th terms of the
arithmetic sequence with the first term is 3
and the common difference is 4.
Solution:
𝑎1 = 3,
𝑎𝑛 = 𝑎1 +
𝑑 = 4
𝑛 − 1 𝑑
𝑎5 = 3 + 5 − 14 = 3 + 16 = 19
𝑎11 = 3 + 11 − 14 = 3 + 40 = 43
Therefore, 19 and 43 are the 5th and the
11th terms of the sequence,respectively.

4. Give the common difference and find the
indicated term in each arithmetic sequence.
1. 1, 5, 9, 13,.. (a10 )
2. 13, 9, 5, 1,… (a10 )
3. -8, -5, -2, 1, 4,.. (a12 )
4. 5, 9, 13, 17,… (a15 )
5. 2, 6, 10,…(a6 )
6. 2, 11, 20, … (a7 )
7. 9, 6, 3,… (a8 )

5. Answer the following:
1. Find the 11th term of the arithmetic
sequence 3,4,5,…
2. Find the 20th term of the arithmetic
sequence 17, 13, 9,…
3. Find the 42nd term of the sequence
5,10,15,..
4. If a1 =5, an =395, and d=5, find the value
of n.
5. If a1 = 5 and a7 = 17, find the common
difference.

6. 6.The 4th term of an arithmetic sequence is
18 and the sixth term is 28. Give the first 3
terms.
7.Write the third and fifth terms of an
arithmetic sequence whose fourth term is 9
and the common difference is 2.
8.Write the first three terms of an arithmetic
sequence if the fourth term is 10 and d = -3

7. Solve the ff.
1. 2, 6, 10,…(a6 )
2. 2,11, 20, … (a7 )
3. 9,6,3,… (a8 )
4. Find the 42nd term of the sequence
5,10,15,..
5. If a1 =5, an =395, and d=5, find the value
of n.

8. ARITHMETIC SEQUENCE

9. A sequence is arithmetic if the differences
between consecutive terms are the same.
9 – 4 = 5
14 – 9 = 5
19 – 14 = 5
24 – 19 = 5
4, 9, 14, 19, 24, . . .
arithmetic
sequence
The common difference, d, is
5.
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10. A sequence is arithmetic if the differences
between consecutive terms are the same.
9 – 4 = 5
14 – 9 = 5
19 – 14 = 5
24 – 19 = 5
4, 9, 14, 19, 24, . . .
arithmetic
sequence
The common difference, d, is
5.
© iTutor. 2000-2013. All Rights Reserved

11. This is an arithmetic sequence.
d = 4
Example:Find the first five terms of the
sequence and determine if it is arithmetic.
an = 1 + (n – 1)4
a1 = 1 + (1 – 1)4 = 1 + 0 = 1
a2 = 1 + (2 – 1)4 = 1 + 4 = 5
a3 = 1 + (3 – 1)4 = 1 + 8 = 9
a4 = 1 + (4 – 1)4 = 1 + 12 = 13
a5 = 1 + (5 – 1)4 = 1 + 16 = 17

12. What is the nth term for each sequence
below.
1. 1,5,9,13,…
2. 13, 9,5,1…
3. -7, -4, -1,2,..
4. 5,3,1,-1,-3
5. 2,6,10,..

13. The nth term of an arithmetic sequence has
the form
an = dn + c
where d is the common difference and c = a1 – d.
a1 =2 c = 2 – 6 = – 4
2, 8, 14, 20, 26, . . . .
d = 8 – 2 = 6
The nth term is 6n – 4.
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14. Example: 1. Find the formula for the nth term
of an arithmetic sequence whose common
difference is 4 and whose first term is 15.
Find the first five terms of the sequence.
an = dn + c
a1 – d =15 – 4 = 11
= 4n + 11
The first five terms are
a1 = 15
15,19, 23, 27, 31.
d = 4
© iTutor. 2000-2013. All Rights Reserved

15. 2.Find the formula for the nth term of an
arithmetic sequence whose common difference
is 3 and whose first term is 5. Find the first five
terms of the sequence.
3.The first term of an arithmetic sequence is
equal to 6 and the common difference is equal
to 3. Find a formula for the nth term of an
arithmetic sequence.
4.Find the formula for the nth term of an
arithmetic sequence whose common difference
is -18 and whose first term is 7. Find the first
five terms of the sequence.

16. Test Yourself:
1. Find the formula for the nth term of an
arithmetic sequence whose common
difference is 15 and whose first term is 3.
Find the first five terms of the sequence.
2. The first term of an arithmetic sequence is 5
and the common difference is 5, find the nth
term of the sequence and its first 6th terms.

17. Try this:
Find the nth term of the ff. sequence.
1. 17,13,9,… d= -4
2. 5,10,15,… d= 5
3. 2,11,20,.. d= 9
4. 9,6,3,… d= -3
5. 5,9,13,17,.. d= 4

18. Assignment:
1. What is arithmetic mean?

19. ARITHMETIC MEAN

20. 1. Find three terms between 2 and 34 of an
arithmetic sequence.
Guide Question:
1. Were you able to get the 3 terms in each
sequence?

21. Arithmetic Mean
 The terms between 𝑎1 and 𝑎𝑛 of an arithmetic
sequence are called arithmetic means of
𝑎1 and 𝑎𝑛. Thus, the arithmetic means
between 𝑎1 and 𝑎5 are 𝑎2, 𝑎3 and
• 𝑎4
 The arithmetic mean
between two numbers
or the “mean”
is sometimes
called the average of two numbers.

22. Sample Problem
1. Find four arithmetic means between 8 and -7.
Answer: Since we must insert four numbers between 8 and -7,
there are six numbers in the arithmetic sequence. Thus, 𝑎1 = 8
and 𝑎6 = −7, we can solve for 𝑑 using the formula
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑.
−7 = 8 + 6 − 1 𝑑
𝑑 = −3
Hence, 𝑎2 = 𝑎1 + 𝑑 = 8 − 3 = 5
𝑎3 = 𝑎2 + 𝑑 = 5 − 3 = 2
𝑎4 = 𝑎3 + 𝑑 = 2 − 3 = −1
𝑎5 = 𝑎4 + 𝑑 = −1 − 3 = −4
Therefore, the four arithmetic means between 8 and -7
are 5, 2, -1, and -4.

23. TEST YOURSELF
1. Insert seven arithmetic means between 3 and
23.
2. Insert four arithmetic means between 8
and 18.
3. Insert six arithmetic means between 16
and 2.
4. Insert five arithmetic means between 0
and -12.
5. Insert 5 arithmetic means between 7 and 70.

24. EXAMPLES
1. Insert 4 arithmetic means between 5 and
25.
2. What is the arithmetic mean between 27
and -3?
3. Insert three arithmetic means between 2
and 14.
4. Insert eight arithmetic means between 47
and 2.

25. Summing Up
What is the sum of the terms of each finite
sequence below?
1. 1,4,7,10
2. 3,5,7,9,11
3. 10,5,0,-5,-10,-15
4. 81,64,47,30,13,-4
5. -2,-5,-8,-11-14,-17
22
35
-15
231
-57

26. 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
know how he did it? Let us find out by doing
the activity below.

27. Determine the answer to the above problem.
Discuss your technique (if any) in getting the
answer quickly. Then answer the question
below.
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?

28. Arithmetic Sum
𝑆𝑛 = n 𝑎1 + (𝑎1+ 𝑛 − 1
𝑑)
2
or
𝑛
S = 𝑛
2
2𝑎1 + 𝑛 − 1
𝑑

29. Examples:
1. Find the sum of the first 10 terms of the
arithmetic sequence 5, 9, 13, 17,…
2. Find the sum of the first 20 terms 20
terms of the arithmetic sequence -2, -5, -
8, -11,…
3. Find the sum of the first ten terms of the
arithmetic sequence 4, 10, 16,…

30. 4.How many terms of the arithmetic
sequence 20, 18, 16,… must be added so
that the sum will be -100?
Therefore, the first 25 terms of the
sequence 20,18,16,… must be added to
get sum of -100.
5. Find the sum of integers from 1 to 50.
6.Find the sum of odd integers from 1 to
100
7.Find the sum of even integers from 1 to
101.

31. Test Yourself
Find the sum of the arithmetic sequence
wherein:
1. 𝑎1= 2; d=4, n=10
2. 𝑎1=10; d= -4; n=8
3. 𝑎1= -7; n=18; d= 8
4.Find the sum of multiples of 3 between 15
and 45.
5.Find the sum of the first eight terms of the
arithmetic sequence 5, 7, 9,…