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Question
Find the sum of all three digit natural numbers, which are multiples of 7 ?
Solution
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 nth term of A.P.
∴ an = 994
⇒ 105 + (n − 1) × 7 = 994 [ `therefore`an = 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.
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