Everything Maths Grade 12

3.2 CHAPTER 3. SEQUENCES AND SERIES

5. 3; 0;− 3;− 6;− 9;− 12;...

General Equation for the n

th

-Term of an

Arithmetic Sequence

EMCP

More formally, the number we start out with iscalled a 1 (the first term), and thedifference between

each successive term isdenoted d, called the common difference.

The general arithmetic sequence looks like:

a 1 = a 1

a 2 = a 1 +d

a 3 = a 2 +d = (a 1 +d) +d = a 1 + 2d

a 4 = a 3 +d = (a 1 + 2d) +d = a 1 + 3d

...

an = a 1 +d. (n− 1)

Thus, the equation for the nth-term is:

an= a 1 +d. (n− 1) (3.1)

Givena 1 and the common difference, d, the entire set of numbers belonging to an arithmetic sequence

can be generated.

DEFINITION: Arithmetic Sequence

An arithmetic (or linear) sequence is a sequence of numbers in which each new term

is calculated by addinga constant value to the previous term:

an= an− 1 +d (3.2)

where

• anrepresents the new term, the nth-term, that is calculated;


• an− 1 represents the previousterm, the (n− 1)th-term;


• d represents some constant.
  Tip

Test for Arithmetic Se-
quences A simple test for an arithmetic sequence is to check that the differencebetween consecutive terms is
constant:

a 2 −a 1 = a 3 −a 2 = an−an− 1 = d (3.3)

This is quite an important equation, and is the definitive test for an arithmetic sequence. If this condi-

tion does not hold, the sequence is not an arithmetic sequence.