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Browse Classroom Capsules and Notes

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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.

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