Q13 of 46 Page 1

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

Listing all 3-digit natural numbers which are multiples of 7 are –

105, 112, 119, … , 994


We have to find 105 + 112 + 119 + … + 994


This is clearly an AP.


Here, a = 105, d = 7 and an = 994 …(i)


Lets find n.


a + (n – 1)d = an …(ii)


Substituting equation (i) in (ii),


105 + (n – 1)×7 = 994


⇒ 7n – 7 = 994 – 105 = 889


⇒ 7n = 889 + 7 = 896


⇒ n = 896/7 = 128


There are 128 (=n) numbers in the series.


Now sum of this AP is given by


S128 = (n/2)[a + an]


⇒ S128 = (128/2)[105 + 994]


⇒ S128 = 64 × 1099 = 70336


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


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