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Find the sum of integers between $100$ and $200$ which are (i) divisible by $9$ (ii) not divisible by $9$.
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(i) Integers divisible by $9$ are $108, 117, 126, ....., 198$
a = $108$, d = $9$
a + $(n-1)d = 198$
$\Rightarrow 108 + (n - 1)9 = 198 \Rightarrow n = 11$
$S_{11} = \frac{n}{2}(a+l) = \frac{11}{2}(108 + 198)$
$= 1683$
(ii) Sum of all integers = $\frac{99}{2}(101 + 199) = \frac{99}{2} \times 300 = 14850$
Sum of integers not divisible by $9 = 14850 - 1683 = 13167$
a = $108$, d = $9$
a + $(n-1)d = 198$
$\Rightarrow 108 + (n - 1)9 = 198 \Rightarrow n = 11$
$S_{11} = \frac{n}{2}(a+l) = \frac{11}{2}(108 + 198)$
$= 1683$
(ii) Sum of all integers = $\frac{99}{2}(101 + 199) = \frac{99}{2} \times 300 = 14850$
Sum of integers not divisible by $9 = 14850 - 1683 = 13167$