For notation purposes I will let the first element of each sum be o1 and e1, the second o2 and e2 and so on.
You have probably noted that for each pair o1 and e1 or o2 and e2 we always have
o1=3 < 4 = e1 hence o1 < e1
o2 = 5 < 6 = e2 hence o2 < e2
…..
Hence O = 3 + 5 + 7 + …. + 103 < 4 + 6 + 8 + … + 104 = E
Now how much smaller is O than E?
Well, you probably noted that each of the pairs I talked about above were always differing by ONE. So if we know the number of pairs, we know the total difference.
Now, to compute the number of pairs we will first calculate the number of even and odd numbers between 1 and 100. as there is one odd mumber for each even one, we have exactly 50 of each.
Now, lets take a look at O for example. It does not include 1 but 101 and 103, so it has 50 - 1 + 2 = 51 elements in it sum.
Therefore there are 51 pairs, and therefore the difference between O and E is 51