# My son’s Sum of Some is beautiful! But what is the proof or explanation?

My youngest son is in $$66$$th grade. He likes to play with numbers. Today, he showed me his latest finding. I call it his “Sum of Some” because he adds up some selected numbers from a series of numbers, and the sum equals a later number in that same series. I have translated his finding into the following equation:
$$(100×2n)+(10×2n+1)+2n+3=2n+7.(100\times2^n)+(10\times2^{n+1})+2^{n+3}=2^{n+7}.$$

Why is this so? What is the proof or explanation? Is it true for any $$nn$$?

His own presentation of his finding:

Every one of these numbers is two times the number before it.
$$1,2,4,8,16,32,64,128,256,512,1024,2048,4096,81921, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192$$.
I pick any one of them, times $$100100$$. Then I add the next one, times $$1010$$. Then I skip the next one. Then I add the one after that.
If I then skip three ones and read the fourth, that one equals my sum!

Factor out the $2^n$ and you get: $2^n (100+20+8) = 2^n 128 = 2^{n+7}$ since $2^7 = 128$