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*