Proofs without words
The following demonstrate proofs of various identities and theorems using pictures, inspired from this gallery.
Summations
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]
Nichomauss' Theorem: can be written as the sum of consecutive integers, and consequently that .
Another proof of the identity .
The identity , where is the th Fibonacci number.
Geometric series
The infinite geometric series .
The infinite geometric series .
The infinite geometric series .
Another proof of the identity .
The infinite geometric series .
The arithmetic-geometric series , also known as Gabriel's staircase.[2]
Geometry
First of several proofs of the Pythagorean Theorem: .[3]
The smallest distance necessary to travel between , the x-axis, and then for is given by .
Another proof of the Pythagorean Theorem (animated version).
In trapezoid with , then .
Miscellaneous
The Root-Mean Square-Arithmetic Mean-Geometric Mean inequality, .
The Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality.[5]
Fermat's Little Theorem: for (above ).
References
- ^ MathOverflow
- ^ Wolfram MathWorld
- ^ Attributed to the Chinese text Zhou Bi Suan Jing.
- ^ This is more of a proof without words of the AM-GM inequality ; though the lengths of the segments labeled RMS and HM can easily be verified to have values of , respectively, it might not be obvious from the diagram. It still serves as a useful graphical demonstration of the inequality.