Algebra Honors ( Periods 4 & 7)

Chapter 3 -5 Arithmetic Sequences as Linear Functions

An arithmetic sequence is an ordered list of numbers ( called terms) where there is a common difference ( d) between consecutive terms. the common difference can either be positive ( increasing)  or negative ( decreasing)

Because an arithmetic sequence has a constant difference, it is a linear function. There is a formula using this common difference to find the equation of any arithmetic sequence:

a_n = a₁ + ( n-1)d

Notice that it is saying that any term in the sequence, a_n,  can be found by adding 1 less than the number of terms of the common difference to the first term, a_1.   Why 1 less than the number of the terms you want?

Take the following arithmetic sequence:

…17, 21, 25,29, 33….

If it’s the 2^nd term, there is only 1 common difference of 4 between the 2 terms. If it is the 3^rd term, there would be 2 common differences of 4 between the 3  terms, etc.

n is always positive because it represents the number of terms and that can’t be negative.

Where  a_n  represents any term in the sequence and a₁ represents the first term in a sequence, n represents the number of the terms in a sequence and d represents the common difference between consecutive terms in a sequence.

This is called an EXPLICIT FORMULA You can explicitly find any term number in the sequence as long as you know the 1^st term and the common difference. For example if the the common difference is 4 and the first term is 1 and you are trying to find the 27^th term

a_n = a₁ + ( n-1)d

a₂₇ = 1 + ( 27-1)4

a₂₇ = 1 + ( 26)4

a₂₇ = 1 + ( 104

a₂₇ = 105

You can also find the next term in the sequence if you know the
RECURSIVE FORUMULA

This is a Function Rule that tells you what the relationship between consecutive terms is. For example if the common difference is 4 and your last term is 101 the next term is 105—without knowing any of the other preceding terms. The terms in the sequence are shown as a list with 3 periods (called an ellipsis) at the end showing that it continues infinitely.

For example, the arithmetic sequences for the above example was

1, 5,9, 13, …

Graphing the terms of an arithmetic sequence shows that it is a linear function

a_n = 1 + ( n-1)4

Simplify

a_n = 1 + ( n-1)4

a_n = 1 + 4n -4

a_n = 4n -3

Now just substitute y for a_n   and x for n

y = 4x – 3

Notice that d is now the slope and also notice that the domain of the sequence is the natural numbers (the counting numbers) because you can’t have a negative term number!

Graph 3 points using 1, 2, 3 for the first 3 terms

(1, 1) ( 2,5) (3, 9)