See a pattern? It looks like the following formula holds for integers :

There is an intuitive geometric explanation for this formula. Consider arranging an by square lattice of beads. There is a total of beads in this lattice. We can imagine many different ways to walk through the lattice and count them; all such ways will lead to the answer . One way to walk through the lattice is to start at a corner and count along the diagonals. If we walk through the entire square lattice of beads like this, for , we would count â€“ where the contribution comes from the main diagonal and the contributions come from the two opposing cornersâ€“ and get since this is the total number of beads.

Applied Mathematician Steven Strogatz posted a photo on Twitter that illustrates this geometric interpretation of the formula . Each type of bead classifies a diagonal on the square lattice of beads.

Proof by beads: 1+2+3+4+3+2+1 = 4^2. Same idea with square of side n shows 1+2+...+ n + ... +2+1 =n^2. pic.twitter.com/uxKGxwQGAW

— Steven Strogatz (@stevenstrogatz) February 22, 2015

The more rigorous way to prove

is by Mathematical induction.

The base case is . Of course, , i.e., is true.

The inductive step is to assume that holds and then show that this implies .

Somehow, we need to construct on the right-hand side of . Noting that , we can add to both sides

The left-hand side can be rewritten by noting that :

Can you recognize this as ? We have just proven that starting with . That is, we have proven:

completing the proof by induction.

The geometric interpretation is much nicer right?

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