Question

Find the sum of all odd natural numbers less than $200$.

• A. $1000$
• B. $10100$
• C. $9900$
• D. $10000$

Answer 1

The correct option is D $10000$

The odd natural numbers less than 200 are $1,3,5...199$

The numbers form an AP

Where, first term $a=1$

and common difference $d=2$

and last term $a_n=199$

We know that $n^{th}$ term of an AP $=a+(n-1)d$

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

$\Rightarrow \frac{198}{2}=n-1$

$\Rightarrow n=99+1$

$\Rightarrow n=100$

We know

Sum of $n$ terms of an AP $=\frac{n}{2}(2a+(n-1\left)d\right)$

Let the sum of odd natural numbers less than $200$ be $S_{100}$

$\Rightarrow S_{100}=\frac{100}{2}[1+199]$

$\Rightarrow S_{100}=50\times 200$

$=10000$