Arithmetic Sequences 9.4A

Infinite SequenceAn infinite sequence is a function whose

domain is the set of positive integers.

Ex. f(n) = 5n + 1

n f(n)

1 6

2 11

3 16

4 21

5 26

Terms of a SequenceThe letter ‘a’ is used to represent sequential functions. The functional value of ‘a’ at ‘n’ is written an (read as “a

sub n”).The sequence is expressed as a1, a2, a3, a4, etc.

Ex. Find the first five terms of the sequence an = 2n – 3

a1 = 2(1) – 3 = -1

a2 = 2(2) – 3 = 1

a3 = 2(3) – 3 = 3

a4 = 2(4) – 3 = 5

a5 = 2(5) – 3 = 7

Arithmetic SequenceAn arithmetic sequence is a sequence that has a common

difference between successive terms.Ex. 1, 8, 15, 22, 29 common difference = 7Ex. 4, 7, 10, 13, 16 common difference = 3

a1, a2, a3, a4… is an arithmetic sequence if and only if there is a real number ‘d’ such that ak+1 – ak = d for every positive integer k, where d is the common difference.

First term: a1

Second term: a1 + d

Third term: a1 + 2d

Fourth term: a1 + 3d

nth term: a1 + (n – 1)d

The General Term of an Arithmetic Sequence The General Term of an Arithmetic Sequence is

given by an = a1 + (n – 1)d

Ex. 6, 2, -2, -6The common difference is -4, so d = -4The First Term is 6, so a1 = 6

an = a1 + (n – 1)d

an = 6 + (n – 1)(-4)

an = 6 – 4n + 4

an = -4n + 10

Find a specific term of a sequenceEx. Find the 40th term of the arithmetic sequence 1, 5,

9, 13…

Common difference = 4First term = 1

an = a1 + (n – 1)d

an = 1 + (n – 1)(4)

an = 1 + 4n – 4

an = 4n – 3

a40 = 4(40) – 3 = 160 – 3 = 157

Find the first term of a sequenceFind the first term of an arithmetic sequence

where the fourth term is 26 and the ninth term is 61.

26 = a1 + (4 – 1)d = a1 + 3d

61 = a1 + (9 – 1)d = a1 + 8d

a1 + 3d = 26 a1 + 3(7) = 26

a1 + 8d = 61 a1 + 21 = 26

-5d = -35 a1 = 5

d = 7