What is the sum of all the odd integers between 500 and 700?

60000

We need to  solve this equation for  n  and then add 1 to the answer to findthe number of terms

501 + 2n  =699

2n =  699 - 501

2n = 198

n = 99

Number of terms  = 99 + 1   =100

Sum =   [ first term + last term ] [ number of terms / 2 ]

[ 501 + 699 ]  [ 100/2]  =

1200 * 50  =

60000

\(501+503+\dots+697+699\\~\\ =\quad (501+2\cdot0) + (501+2\cdot1) +\dots+(501+2\cdot98) +(501+2\cdot99) \\~\\ =\quad (499+2\cdot1) + (499+2\cdot2) +\dots+(499+2\cdot99) +(499+2\cdot100) \\~\\ =\quad \sum\limits_{n=1}^{100}(499+2n) \\~\\ =\quad \sum\limits_{n=1}^{100}499+ \sum\limits_{n=1}^{100}2n \\~\\ =\quad 100\cdot499+ 2\cdot\sum\limits_{n=1}^{100}n \\~\\ =\quad 49900+ 2\cdot\frac{100(100+1)}{2} \\~\\ =\quad 49900+ 10100\\~\\ =\quad60000\).

Thanks Everyone!!! :)