The document defines sequences and series, and discusses arithmetic sequences in particular. An arithmetic sequence is a sequence where each term after the first is obtained by adding a fixed number, called the common difference, to the preceding term. The document provides examples of finding the common difference using the formulas d=an+1-an and d=an-a1. It also gives examples of finding specific terms of an arithmetic sequence given information like the first term, common difference, and nth term.

2. Sequence
• Is a set of numbers that is arranged in
a definite order, each number is
called Term.
Example:
1, 2, 3, 4, 5, …
1st Term
2nd Term
3rd Term
5th Term
4th Term
Three dots means
Goes on forever
(Infinite)

3. Series
• is the sum of terms in the sequence.
Example:
1 + 2 + 3 + 4 + 5 = 15
“TERMS”
“Sum of the
Terms”

5. Arithmetic Sequence
• a sequence of numbers in which each
term after the first is obtained by adding a
fixed number to the preceding term. The
fixed term is called the common difference
(d). The common difference is obtained by
using the formula, d=an+1-an, where n is a
natural number.

6. Arithmetic Sequence
Generic Sequence
Corresponds to the term number
• Note: d=a n +1-an can be used if there is
other terms aside from the first and last
term.
=n is any positive integer greater than 1.

7. Arithmetic Sequence
Example :
5, 10, 15, 20, 25 is an arithmetic sequence.
What is the common difference of this sequence?
The given formula : d=an+1-an

8. Arithmetic Sequence
Example 1:
5, 10, 15, 20, 25
Using the formula:
d=a n +1-an
Given :
d = ?
a1 = 5
a1+1 = 10
Substitute:
d= 10 – 5
d= 5
+ 5 + 5 + 5 + 5

9. Arithmetic Sequence
• Take note that there is another way or
formula to get the common difference of
an arithmetic sequence. If the given is the
first and last term only we can use this
formula, d=an-a1.
n-1

10. Arithmetic Sequence
• Formula: d=an-a1.
n-1
• where:
d- common difference
a1-the first element or term
an-nth element or last term
n-the number of terms

11. Arithmetic Sequence
Example:
What is the common difference (d) if the first term
(a1) of the arithmetic sequence is 11 and the 9th
term is 67?
d=an-a1
n-1
=67-11
9-1
= 56
8 d= 7

12. Arithmetic Sequence
Example #2 :
What is the common difference of an
arithmetic sequence whose first term is -8
and the 7th term is 22?

13. Answer:
d=an-a1
n-1
=22-(-8)
7-1
=30
6
d= 5

14. Derivation Of Formula
n-1 d= an-a1
n-1

15. Example :
What is the 15th term of an arithmetic sequence
-11, -6, -1?
Answer :
d=an+1-an
d=-6-(-11)
d=5
an=a1+(n-1)d
a15=-11+(15-1)5
a15=-11+(14)5
a15=-11+70
a15=59

16. Example :
Find the first term of an arithmetic sequence whose
5th term is 49 and the common difference is 13.
Answer :
a1=an-(n-1)d
=49-(5-1)13
=49-(4)13
=49-52
a1=-3

17. Sol LeWitt’s sculpture of the
four sided pyramid

18. Sol LeWitt’s sculpture of the
four sided pyramid

19. Sol LeWitt’s sculpture of the
four sided pyramid

20. Sol LeWitt’s sculpture of the
four sided pyramid
* a1=4 (the first layer is compose of four
concrete blocks)
*d=8 (every layer of the pyramid increases of 8
concrete blocks)
* an=a1+(n-1)d
a12=4+(12-1)8
a12=4+(11)8
a12=4+88
a12=92 (the 12th layer of the pyramid is
compose of 92 concrete blocks)

21. Other Figures

22. Other Figures

23. Application
• Determine if the following sequence is an arithmetic sequence or
not. Put a if it is an arithmetic sequence and x if not.
1) 7, 11, 15, 19, 23…
2) 2, 8, 32, 128…
3) 20, 18, 16, 14…
4) -11, -10, -9, -8…
5) 1, 2, 4, 8, 16…
6) 4, 8, 12, 16, 20…
7) 1, 3, 9, 27…
8) 5, 8, 11, 14…
9) 1, 3, 5, 7, 9, 11…
10) 3, 6, 12…

24. Assignment
1. Find the common difference of an arithmetic
sequence whose first term is 23 and the
twenty-first term is -17.
2. Find the 80th term of the arithmetic sequence
5,8,11, 14…
3. Find the definition of arithmetic series.

25. A thought for today….

26. TEACHERS OF THE DAY 