Question

Find the sum of the integers between 100 and 200 that are

1. divisible by 9
2. not divisible by 9

[Hint (ii) : These numbers will be : Total numbers – Total numbers divisible by 9]

Answer 1

i. 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.

ii. The sum of the integers between 100 and 200 which is not divisible by 9 = (sum of total numbers between 100 and 200) – (sum of total numbers between 100 and 200 which is divisible by 9)    ...(i)

Total numbers between 100 and 200 is 101, 102, 103,..., 199

Here, a = 101, d = 102 – 101 = 1 and a_n = l = 199

`\implies` 199 = 101 + (n – 1)1    ...[∵ a_n = l = a + (n – 1)d]

`\implies` (n – 1) = 98

`\implies` n = 99

Sum of terms between 100 and 200,

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

`\implies` S₉₉ = `99/2[2(101) + (99 - 1)1]`

= `99/2[202 + 98]`

= `99/2 xx 300`

= 99 × 150

= 14850

From equation (i), sum of the integers between 100 and 200 which is not divisible by 9

= 14850 – 1683    ...[From part (i)]

= 13167

Hence, the required sum is 13167.