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(B) The sum of integers between $100$ and $200$ not divisible by $9$ is calculated as: (Sum of all integers between $100$ and $200$) $-$ (Sum of integ

Vedclass · Accepted Answer

$13167$

Vedclass · Answer

(B) The sum of integers between $100$ and $200$ not divisible by $9$ is calculated as: (Sum of all integers between $100$ and $200$) $-$ (Sum of integers between $100$ and $200$ divisible by $9$).Step $1$: Sum of all integers between $100$ and $200$ (i.e., $101, 102, \dots, 199$).Here, $a = 101$, $l = 199$, and $d = 1$.Number of terms $n = 199 - 101 + 1 = 99$.Sum $S_{99} = \frac{n}{2}(a + l) = \frac{99}{2}(101 + 199) = \frac{99}{2}(300) = 99 	imes 150 = 14850$.Step $2$: Sum of integers between $100$ and $200$ divisible by $9$.The first multiple of $9$ after $100$ is $108$ $(9 	imes 12)$ and the last is $198$ $(9 	imes 22)$.This forms an Arithmetic Progression: $108, 117, \dots, 198$.Here, $a = 108$, $l = 198$, $d = 9$.$198 = 108 + (n-1)9 \Rightarrow 90 = (n-1)9 \Rightarrow n-1 = 10 \Rightarrow n = 11$.Sum $S_{11} = \frac{11}{2}(108 + 198) = \frac{11}{2}(306) = 11 	imes 153 = 1683$.Step $3$: Required sum $= 14850 - 1683 = 13167$.

Find the sum of the integers between $100$ and $200$ that are not divisible by $9$.

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