SEQUENCE

A sequence is a function whose

domain is the finite set {1,23, …, n} or

the infinite set {1,2 ,3,…}.

Example:

n 1 2 3 4 5

a(n) 3 4 5 6 7

Example:

n 1 2 3 4 …

a(n) 2 4 6 8 …

Activity: Term after TermFind the first five

terms of the sequence given the nth term.

Activity: Getting to Know YouWhat is the nth

term for each sequence below?

1. 3, 5, 7, 9, 11, …

2. 3, 4, 5, 6, 7, …3. 2, 4, 8, 16, 32, …

4. -1, 1, -1, 1, -1, …

5. 1, ½ , 1/3, ¼, 1/5,…

ARITHMETIC SEQUENCE

An arithmetic sequence is a sequence where every term after the first is obtained by adding a constant called the common difference.

The sequences 1,4,7,10,... and 15, 11, 7, 3,... are examples of arithmetic sequences since each one has a common difference of 3 and -4.

Arithmetic Rule

an= a1+(n - 1)d•a1 is the first term in the sequence•n is the number of the term you are trying to determine•d is the common difference•an is the value of the term that are looking for

Try this…Use the arithmetic

formula to determine the 100th term of the following sequence:

75, 25, -25, -75, -125,…

an = a1 + (n - 1)d = 75 + (100 – 1)(-50)

= -4875a1 = 75n = 100d = -50

Try this…Use the arithmetic

formula to determine the 10th term of the following sequence:

5, 12, 19, 26,…

an = a1 + (n - 1)d = 5 + (10 – 1)(7)

= 68a1 = 5n = 10d = 7

What is missing?

1. 3, 12, 21, __ , __ , __30 39 48

2. 8, 3, -2, __ , __-7 -12

3. 5, 12, __ , 26, __ 19 33

What is missing?

4. __, __,__, 8 , 12 , 16-4 0 4

5. -1, __ , __,__, 31, 39 7 15

6. 13,__ ,__,__,-11.-17 7 123

-5

Insert three terms between 2 and 34 of an

arithmetic sequence.

ARITHMETIC MEANS

The terms between any two consecutive terms of an arithmetic sequence.

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 and 5 ,𝑎2 ,𝑎3 ,𝑎4 ,𝑎5 ,25

Let 5 ,𝑎2 ,𝑎3 ,𝑎4 ,𝑎5 ,25

25=5+ (5 ) 𝑑20−5=5𝑑4=𝑑

5 ,𝑎2 ,𝑎3 ,𝑎4 ,𝑎5 ,25𝑎2=𝑎1+(1)(𝑑 ) n = 2

(2 – 1) = 1𝑎2=5+(1 ) (4 )=9=13

𝑎4=5+ (3 ) (4 )=17𝑎5=5+(4 ) (4 )=21

Insert 2 arithmetic means between ½ and 2.

ARITHMETIC SUMSum of the terms in

an arithmetic sequence.

What is 1 + 2 + 3 + …+ 50 +

51 +… + 98 + 99 +100?

Formula: 𝑎𝑛=𝑎1+ (𝑛−1 ) 𝑑

Example:

Find the sum of the first 10 terms of the arithmetic sequence 5, 9, 13, 17,…

Using the formula:

𝑆10=102

[2(5)+(10−1 ) 4]

d = 9 - 5

𝑆10=5 [10+ (9 )4 ]=5 (46 )=𝟐𝟑𝟎

Example:

Find the sum of the first 20 terms of the arithmetic

sequence -2, -5, -8, -11, …

- 610

Activity: How well do

you know me?

Which of the following sequences is an

arithmetic sequence? Why?

1. 3, 7, 11, 15, 192. 4, 16, 64, 2563. 48, 24, 12, 6, 3, …

4. 1, 4, 9, 16, 25, 365. 1, ½ , 0, - ½

6. – 2, 4, - 8, 16, …7. 1, 0, -1, -2, -38. ½ , 1/3, ¼ , 1/59. 3x, x, x/3, x/910. 9.5, 7.5, 5.5, 3.5

Find the 25th term of the arithmetic

sequence 3, 7, 11, 15, 19,…

The second term of the arithmetic

sequence is 24 and the fifth term is 3.

Find the first term and the common

difference

Give the arithmetic

sequence of 5 terms if the first term is 8 and the last term is 100.

What must be the value of k so that 5k – 3, k +2, and 3k – 11 will form

an arithmetic sequence?