Number 3 can be expressed as the sum of one or more positive integers in 4 distinct ways:
3; 2 + 1; 1 + 2; 1 + 1 + 1
Number 4 can be expressed as the sum of one or more positive integers in 8 distinct ways:
4; 3 + 1; 1+3; 2 + 2; 2 + 1 + 1; 1+2+1; 1+1+2; 1+1+1+1
Prove : any positive integer n can be so expressed in 2^{n - 1} ways.

Imagine n 1's in a row.  There are n-1 spaces between them.  In each space either combine or separate.  There are 2^(n-1) ways to choose.

Posted by Jer on 2017-03-26 21:31:02