Question

Find the sum $1+3+5+.....+51$ (the sum of all odd numbers from $1$ to $51$) without actually adding them.

Answer 1

Observe the pattern in sum of consecutive odd numbers :
$1=1^2$
$1+3=4=2^2$
$1+3+5=9=3^2$
$1+3+5+9+7=16=4^2$
$1+3+5+9+7+9=25=5^2$
Hence it can be observed that sum of first $n$ odd number is $n^2$.
In $1$ to $50$ numbers, there are $25$ odd natural numbers and $25$ even numbers.
$51$ is ${26}^{th}$ odd number.
Therefore, $1+3+5+..........+51={26}^2=26\times 26=676$