Question

Find the sum of the integers between 100 and 200 that are divisible by 9

Answer 1

The numbers (integers) between 100 and 200 which is divisible by 9 are 108, 117,126, ..., 198

Let n be the number of terms between 100 and 200 which is divisible by 9.

Here, a = 108, d = 117 – 108 = 9

And a_n = l = 198

`\implies` 198 = 108 + (n – 1)9   ......[∵ a_n = l = a + (n – 1)d]

`\implies` 90 = (n – 1)9

`\implies` n – 1 = 10

`\implies` n = 11

∴ Sum of terms between 100 and 200 which is divisible by 9 is

S_n = `n/2 [2a + (n - 1)d]`

`\implies` S₁₁ = `11/2[2(108) + (11 - 1)9]`

= `11/2[216 + 90]`

= `11/2 xx 306`

= 11 × 153

= 1683

Hence, required sum of the integers between 100 and 200 that are divisible by 9 is 1683.