Find the sum of the numbers of three digits d | vedclass

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Question

(C) The smallest and the largest three-digit numbers divisible by $7$ are $105$ and $994$,respectively.This forms an arithmetic progression: $105, 112

Vedclass · Accepted Answer

$70336$

Vedclass · Answer

(C) The smallest and the largest three-digit numbers divisible by $7$ are $105$ and $994$,respectively.This forms an arithmetic progression: $105, 112, 119, \ldots, 994$.Here,the first term $a = 105$,the common difference $d = 7$,and the last term $a_n = 994$.Using the formula for the $n^{th}$ term: $a_n = a + (n - 1)d$.$994 = 105 + (n - 1) 	imes 7$$994 - 105 = 7(n - 1)$$889 = 7(n - 1)$$n - 1 = 127 \Rightarrow n = 128$.Now,the sum of $n$ terms is given by $S_n = \frac{n}{2}(a + a_n)$.$S_{128} = \frac{128}{2}(105 + 994)$$S_{128} = 64 	imes 1099 = 70336$.

Find the sum of the numbers of three digits divisible by $7$.

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