# Browse Classroom Capsules and Notes

You can filter the list below by selecting a subject category from the drop-down list below, for example by selecting 'One-Variable Calculus'. Then click the 'APPLY' button directly, or select a subcategory to further refine your results.

Displaying 61 - 70 of 133

This article links two seemingly unrelated problems, one in probability and the other in dynamical systems, and shows they are actually one involving Fibonacci numbers.

A geometric demonstration of a recursive identity for triangular numbers

A geometric demonstration of an identity for the sum of products of consecutive integers.

Four geometric demonstrations of an inequality: the sum of a positive number and its reciprocal is at least $2$

A visual solution for an alternating sum of odd squares is presented.

It is known that every positive odd integer can be expressed in the form $x^2 + y^2 + 2z^2$ for some integers $x$, $y$ and $z$. The authors discuss the non-existence of certain...

An odd prime $p$ has $(p-1)/2$ quadratic residues mod $p$, and for relatively prime $p$ and $q$ there are $(p-1)(q-1)/2$ non-representable Frobenius numbers. The author discusses a...

The sum of squares formula is proved visually.
A visual derivation of the formula for the sum of the first $n$ integers is presented.
Six identities, each of which gives infinitely many (but not all) integral solutions to the equation in the title, are shown to be special cases of a more general identity.