NCERT Class 10 Mathematics

ARITHMETIC PROGRESSIONS 103

Example 6 : Check whether 301 is a term of the list of numbers 5, 11, 17, 23,...

Solution : We have :

a 2 – a 1 = 11 – 5 = 6, a 3 – a 2 = 17 – 11 = 6, a 4 – a 3 = 23 – 17 = 6

As ak + 1 – ak is the same for k = 1, 2, 3, etc., the given list of numbers is an AP.

Now, a = 5 and d = 6.

Let 301 be a term, say, the nth term of the this AP.

We know that

an =a + (n – 1) d

So, 301 = 5 + (n – 1) × 6

i.e., 301 = 6n – 1

So, n =

302 151

63

But n should be a positive integer (Why?). So, 301 is not a term of the given list of
numbers.

Example 7 : 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 an =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.

Example 8 : Find the 11th term from the last term (towards the first term) of the

AP : 10, 7, 4,.. ., – 62.

Solution : Here, a = 10,d = 7 – 10 = – 3,l = – 62,

where l = a + (n – 1) d