Question

Find the value of the following:
$21+23+25+\cdots +97+99$

Answer 1

We know that sum of first $n$ odd numbers are $n^2$. So,
Sum of first 50 odd numbers will be,
$1+3+5+7\cdots +97+99={50}^2$
But we have to find the sum of odd numbers starting from 21. So we have to subtract sum of first 10 odd numbers,
Sum of first 10 odd numbers will be
$1+3+5+\cdots +17+19={10}^2$
Now, for our required sum,
$(1+3+5+7\cdots +97+99)-(1+3+5+\cdots +17+19)={50}^2-{10}^2$
$⟹21+23+25+\cdots +97+99={50}^2-{10}^2⟹21+23+25+\cdots +97+99=2500-100⟹21+23+25+\cdots +97+99=2400$