Arithmetic Sequences (Recursive Formulas)

Vocabulary • sequence – a set of numbers in a specific order.

• terms – the numbers in the sequence.

• arithmetic sequence – a numerical pattern that increases or decreases at aconstant rate.

• common difference – the difference between the terms.

What is an arithmetic sequence?

A sequence in which each term is found by Adding or Subtracting the same number to the previous term.

4, 8, 12 , 16, 20…………..

+4 +4+4 +4

What is the common difference?

The difference between each number. This determines what is added to or subtracted from each previous number to obtain the next number.

4, 8, 12, 16, 20………….. 4 is the common difference

Is this an arithmetic sequence?

10, 15, 20, 25, 30……

Yes it is an Arithmetic Sequence with +5 as the

Common Difference

Is this an arithmetic sequence?

2, 9, 17, 29, 37……

No it is NOT an Arithmetic Sequence because there is NO Common Difference

What is the next term in this sequence?

5, 12, 19, 26 , _____

33

+7 +7 +7 +7

What is the next term in this sequence?

5, -1, -7, -13 , _____

-19

-6 -6 -6 -6

What is the next term in this sequence?

100, 75, 50 , 25____

0

-25 -25 -25 -25

Arithmetic Sequence (Recursive) Formula

an = a₁ + (n – 1) d where:

an = nth term (the one you are looking for)

a₁ = 1st term in sequence

n = term number

d = common difference

Used to find terms deep into a sequence (you must have a sequence identified with a common difference)

Find the 9th term of the arithmetic sequence.

7, 11, 15, 19, ...Find the common difference, d: +4 +4 +4 7, 11, 15, 19So… d = 4

Find n: n is the term you are looking for, so… n = 9

Find a₁ (1st term in sequence), so… a₁ = 7

an = a₁ + (n – 1) d

a9 = 7 + (9 – 1) 4

a9 = 7 + (8) 4

a9 = 7 + 32

a9 = 39

Find the 12th term in the arithmetic sequence.

12, 17, 22, 27, ...Find the common difference, d: +5 +5 +5 12, 17, 22, 27So… d = 5

Find n: n is the term you are looking for, so… n = 12

Find a₁ (1st term in sequence), so… a₁ = 12

an = a₁ + (n – 1) d

a12 = 12 + (12 – 1) 5

a12 = 12 + (11) 5

a12 = 12 + 55

a12 = 67

Write an equation for the nth term of the sequence; -8, 1, 10, 19 …

Simplify.

Find the common difference, d: +9 +9 +9 -8, 1, 10, 19So… d = 9

Find a₁ (1st term in sequence), so… a₁ = -8

Now…substitute and simplify the equation

an = a₁ + (n – 1) d

an = -8 + (n – 1) 9 use the distributive property

an = -8 + 9n - 9

an = 9n - 17