Find the number of all multiples of 9 lying between 300 and 700.

Given:

The first multiple of 9 which is greater than 300 is 306 and the next multiple is 315.

∴ 306, 315, 324, 333…

So, first term = a = 306

And common difference = d = 324 – 315 = 9

When 700 is divided by 9, the remainder is 7.

So, 700 – 7 = 693 = l = last term

Then, the series is 306, 315, 324, 333… 693.

Let 693 be the nth term of the A.P.

We know that the nth term or the general term of an A.P is a_n = a + (n – 1) d where a = first term; n = number of terms and d = common difference.

Thus, 693 = 306 + (n – 1) 9

⇒ 693 – 306 = (n – 1) 9

⇒ 387/9 = n – 1

⇒ 43 = n – 1

⇒ 43 + 1 = n

∴ n = 44

Ans. The number of all multiples of 9 lying between 300 and 700 is 44.