First of all, we have that the sum of all the odd integers from #28# to #46# are #29 + 31 + ... + 43 + 45#. That is a total of #9# numbers.
Let's represent this sum as a variable #s#:
#s = 29 + 31 + ... + 43 + 45#
Then reverse the order:
#s = 45 + 43 + ... + 31 + 29#
Add these two each other:
#s + s = (29 + 45) + (31 + 43) + ... + (43 + 31) + (45 + 29)#
#s + s# becomes #2s#. Notice, from the left to right of the first two terms. As #29# increases by #2# to become #31#, we can see that #45# also decreases by #2# to become #43#. Since this goes on all the way, this means that essentisally, all of these terms are the same:
#29 + 45 = 31 + 43 = ... = 43 + 31 = 45 + 29#
And since there are #9# of them, we could just sum one of these terms and multiply by #9#:
#2s = (29 + 45) * 9 = 74 * 9 = 666#.
Finally, we divide both sides by #2#:
#{2s}/2 = 666/2#
#s = 333#
