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

Commom difference (d) = 7

Let 994 be the $n^{th}$ term of A.P.

$a_n=994$

$105+(n-1)\times 7=994[a_n=a+(n-1\left)d\right]$

7(n-1) = 994 - 105

7(n-1) = 889

n-1 = 127

n = 128
sum of all the tems of $A.P.=\frac{128}{2}(105+994)$ $[\because Sn=\frac{n}{2}(a+l),Ibeinglastterm]$

$=64\times 1099$

= 70336

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