#### i)Find the sum of the integers between 100 and 200 that are divisible by 9ii)Find the sum of the integers between 100 and 200 that are not divisible by 9

i)

Numbers between 100 and 200 which is divisible by 9 are 108, 117, 126, ... 198.

Here a = 108, d = 117- 108 = 9

$\\ a\textsubscript{n} = 198\\ a + (n - 1) \times d = 198\\ 108 + (n - 1) \times 9 = 198\\ (n - 1) \times 9 = 198 - 108\\ \left( n-1 \right)=\frac{90}{9} \\$

\\n = 10 + 1 = 11 \\ \mathrm{S}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[2 \mathrm{a}+(\mathrm{n}-1) \mathrm{d}] \\ \text { Put } \mathrm{n}=11 \\ \begin{aligned} \mathrm{S}_{11} &=\frac{11}{2}[2(108)+(11-1) 9] \\ &=\frac{11}{2}[216+90] \\ &=\frac{11}{2} \times 306 \\ &=11 \times 153 \\ \mathrm{~S}_{11} &=1683 \end{aligned}

ii)

Numbers between 100 and 200 are 101, 102, 103,   .. , 199

Here a = 101, d = 102 - 101 = 1

$\\a\textsubscript{n} = 199\\ a + (n - 1).d = 199 ( Using a\textsubscript{n} = a + (n - 1)d )\\ 101 + (n - 1)1 = 199\\ (n - 1) = 199 - 101\\ n = 98 + 1\\ n = 99\\ {{S}_{99}}=\frac{99}{2}\left[ 2\times 101+\left( 99-1 \right)\left( 1 \right) \right] \\$
\begin{aligned} & =\frac{99}{2}\left[ 202+98 \right] \ & =\frac{99}{2}\left[ 300 \right] \ \end{aligned} \\

$\\=99[150]\\ = 14850\\$

Numbers between 100 and 200 which is divisible by 9 are 108, 117, 126,  ...198

Here b = 108,

$d\textsubscript{1} = 117- 108 = 9\\$

$\\ b\textsubscript{N} = 198\\ b + (N - 1) \times d\textsubscript{1 } = 198\\ 108 + (N - 1) \times 9 = 198\\ (N - 1) \times 9 = 198 - 108\\ \left( N-1 \right)=\frac{90}{9} \\$

\\N= 10 + 1 = 11\\ \mathrm{S}_{\mathrm{N}}=\frac{\mathrm{N}}{2}\left[2 \mathrm{~b}+(\mathrm{N}-1) \mathrm{d}_{1}\right] \\ \text { Put } \mathrm{N}=11 \\ \begin{aligned} \mathrm{S}_{11} &=\frac{11}{2}[2(108)+(11-1) 9] \\ &=\frac{11}{2}[216+90] \\ &=\frac{11}{2} \times 306 \\ &=11 \times 153 \\ \mathrm{~S}_{11} &=1683 \end{aligned}

the sum of the integers between 100 and 200 that is not divisible by 9

=Sum of all numbers - the sum of numbers which is divisible by 9

$\\ = 14850 - 1683 \\ = 13167\\$