Question

Find the sum of all three digit natural numbers, which are multiples of 7 ?

Answer 1

Three digit natural numbers which are multiples of 7 are 105, 112, 119,…, 994.

105, 112, 119, … 994 are in A.P.

First term (a) = 105

Common difference (d) = 7

Let 994 be the n^th term of A.P.

∴ a_n = 994

⇒ 105 + (n − 1) × 7 = 994 [ `therefore`a_n = a + (n − 1)d]

⇒ 7 (n − 1) = 994 − 105

⇒ 7 (n − 1) = 889

⇒ n − 1 = 127

⇒ n = 128

`\text{Sumof all terms of AP}=128/2(105+994)[thereforeS_n=n/2(a+l),\text{being last term}]`

`=64xx1099`

`=70336`

Thus, the sum of all three digit natural numbers which are multiples of 7 is 70336.