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a n = a + (n – 1) d.

a n is also called the general term of the AP. If there are m terms in the AP, then a m
represents the last term which is sometimes also denoted by l.

Examples-

Example 1 : Determine the AP whose 3rd term is 5 and the 7th term is 9.

Solution : We have

a 3 = a + (3 – 1) d = a + 2d = 5 (1)

and a 7 = a + (7 – 1) d = a + 6d = 9 (2)

Solving the pair of linear equations (1) and (2), we get

a = 3, d = 1

Hence, the required AP is 3, 4, 5, 6, 7, . . .

Example 2: How many two-digit numbers are divisible by 3?

Solution : The list of two-digit numbers divisible by 3 is :

12, 15, 18, . . . , 99

Is this an AP? Yes it is. Here,

a = 12, d = 3, an = 99.

As a n = a + (n – 1) d,

we have 99 = 12 + (n – 1) × 3

i.e., 87 = (n – 1) × 3

i.e., n – 1 = 87 3 = 29

i.e., n = 29 + 1 = 30

So, there are 30 two-digit numbers divisible by 3.

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