Art of Problem Solving

The following demonstrate proofs of various identities and theorems using pictures, inspired from this gallery. The sum of the first odd natural numbers is . The sum of the first positive integers is . The sum of the first positive integers is .[1] The alternating sum of the first odd natural numbers is . (Source) Nichomauss' Theorem: the sum of the first cubes can be written as the square of the sum of the first integers, a statement that can be written as . Here, we use the same re-arrangement as the first proof on this page (the sum of first odd integers is a square). Here's another re-arrangement to see this: This also suggests the following alternative proof: The th pentagonal number is the sum of and three times the th triangular number. If denotes the th pentagonal number, then . The infinite geometric series . The infinite geometric series . The infinite geometric series . Another proof of the identity . Varignon's theorem: the area of the outer parallelogram is twice

saved by

• Kevin Zhang