Q. 275.0( 2 Votes )

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

Answer :

The three digit natural numbers which are multiples of 7 are

105, 112, 119,…, 994

a_{2} – a_{1} = 112 – 105 = 7

a_{3} – a_{2} = 112 – 105 = 7

∵ a_{3} – a_{2} = a_{2} – a_{1} = 7

Therefore, the series is in AP

Here, a = 105, d = 7 and a_{n} = 994

We know that,

a_{n} = a + (n – 1)d

⇒ 994 = 105 + (n – 1)7

⇒ 994 – 105 = (n – 1)7

⇒ 889 = (n – 1)7

⇒ 127 = (n – 1)

⇒ n = 128

Now, we have to find the sum of this AP

⇒ S_{128} = 64[210 + 127 × 7]

⇒ S_{128} = 64[1099]

⇒ S_{128} = 70336

Hence, the sum of all three digit numbers which are multiples of 7 are 70336.

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