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

My youngest son is in 6th 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.

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

His own presentation of his finding:

Every one of these numbers is two times the number before it.
1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192.
I pick any one of them, times 100. Then I add the next one, times 10. 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!

Answer

Factor out the 2n and you get: 2n(100+20+8)=2n128=2n+7 since 27=128

Attribution
Source : Link , Question Author : haugsire , Answer Author : response

Leave a Comment