If you are not sure how to handle a general problem, it is always a good idea to try simple versions. The sum of the first 5 odd number is 1+ 3+ 5+ 7+ 9= 25. That easier to see if we write
1+ 3+ 5+ 7+ 9 and then
9+ 7+ 5+ 3+ 1 each vertical pair sums to 10 and there are 5 such sums- the total is 5(10)= 50. Since we added twice, the sum is 25.
More generally, if we write
1+ 3+ 5+ ...+n-4+n-2+ n
n+ n-2+n-4+ ...+ 5+ 3+ 1 each vertical pair sums to n+ 1 and there are (n+ 1)/2 terms (remember that n is odd)- the total is \(\displaystyle (n+1)(n+1)/2= (n+1)^2/2\). Since we added twice, the original sum is \(\displaystyle (n+1)^2/4\) (note that, since n is odd, this is an integer).