Question
All integers between $100$ and $550$ which are not divisible by $9$.
✓
Answer
The sum of the integers between $100$ and $550$ which is not divisible by $9$= (Sum of total numbers between $100$ and $550$) - (Sum of totel numbers between $100$ and $550$ which is divisible by $9$)
Here,
$a = 101, d = 102 - 101 = 1$ and $a_n = l = 549$
$\because$ $a_n = l = a + (n - 1)1$
$\Rightarrow 549 = 101 + (n - 1)1$
$\Rightarrow (n - 1) = 448 \Rightarrow n = 449$
$\therefore$ Sum of terms between $100$ and $550$
$\text{S}_\text{n}=\frac{\text{n}}{2}[2\text{a}+(\text{n}-1)\text{d}]$
$\Rightarrow\ \text{S}_{449}=\frac{449}{2}[2\times101+(449-1)1]$
$=\frac{449}{2}[202+448]=\frac{449}{2}\times650$
$= 449 \times325 =145,925$
No. below $100$ and $550$ which are divisible by $9$
$108, 117, 126, 135 ...... 540$
here $a = 108, d = 9, a_n = 540$
Therefore,
$a_n = a + (n - 1)d$
$549 = 108 + (n - 1)9$
$549 = 108 = (n - 1)9$
$=\frac{441}{9}=\text{n}-1$
$49 = n - 1$
$n = 50$
$\text{S}_{50}=\frac{50}{2}(108+549)\Big[\text{S}_\text{n}=\frac{\text{n}}{2}(\text{a}+1)\Big]$
$\text{S}_{50}=\frac{50}{2}(657)$
$\text{S}_{50}=25\times657$
$\text{S}_{50}=16425$
So that from conditior
$= 145,925 - 16,425 = 129, 500$
Hence, the required sum is $129,500.$
Here,
$a = 101, d = 102 - 101 = 1$ and $a_n = l = 549$
$\because$ $a_n = l = a + (n - 1)1$
$\Rightarrow 549 = 101 + (n - 1)1$
$\Rightarrow (n - 1) = 448 \Rightarrow n = 449$
$\therefore$ Sum of terms between $100$ and $550$
$\text{S}_\text{n}=\frac{\text{n}}{2}[2\text{a}+(\text{n}-1)\text{d}]$
$\Rightarrow\ \text{S}_{449}=\frac{449}{2}[2\times101+(449-1)1]$
$=\frac{449}{2}[202+448]=\frac{449}{2}\times650$
$= 449 \times325 =145,925$
No. below $100$ and $550$ which are divisible by $9$
$108, 117, 126, 135 ...... 540$
here $a = 108, d = 9, a_n = 540$
Therefore,
$a_n = a + (n - 1)d$
$549 = 108 + (n - 1)9$
$549 = 108 = (n - 1)9$
$=\frac{441}{9}=\text{n}-1$
$49 = n - 1$
$n = 50$
$\text{S}_{50}=\frac{50}{2}(108+549)\Big[\text{S}_\text{n}=\frac{\text{n}}{2}(\text{a}+1)\Big]$
$\text{S}_{50}=\frac{50}{2}(657)$
$\text{S}_{50}=25\times657$
$\text{S}_{50}=16425$
So that from conditior
$= 145,925 - 16,425 = 129, 500$
Hence, the required sum is $129,500.$
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