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

The three digit natural numbers which are multiples of 7 are

105, 112, 119,…, 994

a2 – a1 = 112 – 105 = 7

a3 – a2 = 112 – 105 = 7

a3 – a2 = a2 – a1 = 7

Therefore, the series is in AP

Here, a = 105, d = 7 and an = 994

We know that,

an = 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  S128 = 64[210 + 127 × 7]

S128 = 64

S128 = 70336

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

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