Find the sum of all odd numbers between 100 and 200.

Given:

Odd numbers between 100 and 200.

To do:

We have to find the sum of all odd numbers between 100 and 200.

Solution:

Odd numbers between 100 and 200 are \( 101,103,105,107, \ldots, 199 \).

The sequence is in A.P.

Here,

\( a=101 \) and \( d=103-101=2 \) \( l=199 \)

We know that,

$l=a+(n-1) d$

$\Rightarrow 199=101+(n-1) \times 2$

$\Rightarrow 199=101+2 n-2$

$\Rightarrow 199-99=2 n$

$\Rightarrow n=\frac{100}{2}=50$

$\therefore n=50$

We know that,

$\mathrm{S}_{n}=\frac{n}{2}[2 a+(n-1) d]$

$=\frac{50}{2}[2 \times 101+(50-1) \times 2]$

$=25[202+49 \times 2]$

$=25(202+98)$

$=25 \times 300$

$=7500$

The sum of all odd numbers between 100 and 200 is $7500$.