# Classroom Capsules and Notes

Capsules By Courses. We are organizing the capsules into courses, when possible using the same topics as are used in Course Communities. So far we have organized capsules for the following courses:

You may select topics within each course.

Notes: Sequences and Series is part of One-Variable Calculus. We felt that since this topic had so many capsules associated with it, we wanted to introduce sub-topics. Also, the Number Theory collection of capsules does not correspond to a course in Course Communities, but has topics selected by the Editorial Board for Classroom Capsules and Notes.

## Featured Items

##### Variations of the Sliding Ladder Problem

This article takes another look at the sliding ladder problem that students meet in the study of related rates in calculus.  Three variations of the problem are analyzed using elementary notions from mechanics, providing interesting insights and resolving an apparent paradox about the speed at which the top of the sliding ladder hits the ground.

##### Sam Loyd's Courier Problem with Diophantus, Pythagoras, and Martin Gardner

In Sam Loyd's classical Courier Problem, a courier goes around an army while both travel at constant speeds. If the army travels its length during the time the courier makes his trip, how far does the courier ride? In both revisions of this problem, a single-file army and a square army, the solution is irrational. Here, variations are considered in which the solutions are rational.   In fact, certain Pythagorean tripes can be used to generate problems that have integer solutions.

##### A Note on Disjoint Covering Systems—Variations on a 2002 AIME Problem

A covering system is a system of $$k$$ arithmetic progressions whose union includes all integers. This paper presents upper bounds on the number of consecutive integers which need to be checked to determine whether a covering system is a disjoint covering system.

##### $$\lim_{m \rightarrow \infty} \sum_{k=0}^{m} (k/m)^m$$ $$= e/(e-1)$$

Two proofs, one elementary, of the limit in the title are presented.

##### Proof without Words: Squares of Triangular Numbers

The author proves visually that the square of the $$n$$th triangular number equals the sum of the cubes of the first $$n$$ positive integers.

##### Does What Goes Up Take the Same Time to Come Down?

A solution is obtained for motion where resistance is proportional to square of velocity.