Let´s consider the set of integers less or equal than a given one:

If we add all the elements in this set, we have:

If we take a look, at the figure, to the right, we can see how the sum ,is equal to the area below the “ladder” plotted:

Where is the area formed for small triangles, with a area, each one; And is the area of an isosceles triangle with an equal base and height of .

And if the aim, of this blog, where to be rigurous instead of, to give simple ideas, an induction proof, should fit here, perfectly [1].

To extend this result to the sum of all the elements, in the subsets included, in the Power Set of integers less or equal to a given one:

There are subsets in , (see [3] and [4]):

If , then this element is in half of the subsets of (observe the relation between the binary digits and the power set [3]):

To get the final result, it is only necessary to multiply both expresions [2]:

**NOTE:**

**Archives:**
[a]-020609-SUMSINSIDEPOWERSET.nb

**References:**
[1]-A000217 The On-Line Encyclopedia of Integer Sequences!

[2]-A001788 The On-Line Encyclopedia of Integer Sequences!

[3]-Powerset Construction Algorithm Shriphani Palakodety.

[4]-Course Notes 8: Size of the Power Set. Chris Nowlin.

### Like this:

Like Loading...

*Related*