Find the sum of the integers between 100 and 200 that are- Divisi… - Vidyadip

Question

Find the sum of the integers between 100 and 200 that are-
Divisible by 9.

✓

Answer

Numbers between 100 - 200 divisible by 9 are 108, 117, 125, ...... 198
Here, a = 108, d = 117 - 108 = 9 and $a_n = 198$
[$\because$ $a_n = a + (n - 1)d]$
$\Rightarrow a + (n - 1)d = 198$
$\Rightarrow 108 + (n - 1)9 = 198$
$\Rightarrow 9[12 + n - 1] = 198$
$\Rightarrow11+\text{n}=\frac{198}{9}$
$\Rightarrow\text{n}=22-11$
$\Rightarrow\text{n}=11$
$\text{Now},\text{S}_{\text{n}}=\frac{\text{n}}{2}\big[2\text{a}+\big(\text{n}-1\big)\text{d}\big]$
$\Rightarrow\text{S}_{11}=\frac{11}{2}\big[2\big(108\big)+\big(11-1\big)\big(9\big)\big]$
$\Rightarrow\text{S}_{11}=\frac{11}{2}\big[216+99-9\big]$
$\Rightarrow\text{S}_{11}=\frac{11}{2}\big[216+90\big]$
$\Rightarrow\text{S}_{11}=\frac{11}{2}\times306$
$\Rightarrow\text{S}_{11}=1683$

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