Question

Study the following pattern and find the sum of first $50$ odd numbers.

$$\begin{gathered} 1^2=1 \\ {2}^2=1+3 \\ {3}^2=1+3+5 \\ {4}^2=1+3+5+7 \end{gathered}$$

Answer 1

If we observe the pattern closely, we will find that the number whose square is on left hand side of the equation is the number of odd terms in the right hand side of the equation as a sum.

Therefore to find the sum of first $50$ odd numbers, we will find out the square of $50$.

For example, $3^2=1+3+5$ (3 terms are in right hand side so we took the square of 3 in left hand side)

Hence, ${50}^2$ will be the sum of first $50$ odd numbers.

Thus, the sum of first $50$ odd numbers $={50}^2=50\times 50=2500$.