Odd natural numbers

Sum of n no. of odd natural numbers is= $n^{2}$

The series of numbers form an arithmetic progression with $a_1=1$ , $d=2$ and $n=10234$ . The $n^{th}$ term is given by

$a_n=a_1+(n-1)d$

$a_{10234}=1+(10233)(2)=20467$

The sum of the terms of an arithmetic progression is given by

$S=\dfrac{n}{2}(a_1+a_n)$ , substitute:

$S=\dfrac{10234}{2}(1+20467)=\boxed{104734756}$

The 10234th positive odd number is 2 × 10234 - 1 = 20467.

We can determine the sum of the first 10234 odd numbers as the difference of:

the sum of the first 20468 positive integers ( $S_{20468}$ )

and

the sum of the first 10234 positive even integers (which is the same as twice the sum of the first 10234 integers, $2 × S_{10234}$ ).

$S_{20468} = \frac {20468 × (20468 + 1)}{2} = 209479746$

$2 × S_{10234} = 2 × \frac {10234 × (10234 + 1)}{2} = 104744990$

$S_{20468} - 2 × S_{10234} = 209479746 - 104744990 = \boxed {104734756}$