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

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21978

22978

21970

21960

answer is A.

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Detailed Solution

Given, range is

300

and

700

Let's write the multiples of 9 as A .P with common difference as 9 .

306, 315, 324, \dots, 693

The first term of AP,

a = 306,

And the common difference,

d = 9

.
Last term of the AP,

l = 639

As we know, n th term of an A .P . is given by the formula as,

a n = a + n − 1 d … 1

Where "a" is the first term of the A .P ., d is the common difference of the A .P ., a n is n th term of the A .P . And n is the total number of terms .

In equation 1, substituting the known values .

⇒ 693 = 306 + n − 1 9

⇒ 693 − 306 = n − 1 9

⇒ 387 = n − 1 9

⇒ n − 1 = 3879

\Rightarrow n - 1 = 43

\Rightarrow n = 44

Between 300 to 700, there are total 44 numbers which are multiples of 9 .

For an A .P . having first term 'a' and last term 'l', sum of the n terms of  is given by,

S n = n 2 a + l

⇒ S 44 = 442 306 + 693

⇒ S 44 = 22999

⇒ S 44 = 21978

Hence, the correct option is 1.

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