Algebra

Homogeneous Linear Recurrence Relations

Last week we looked at a recent question about recurrence relations, and I realized it needs a companion article to introduce these ideas. So here we will look at some answers from Ask Dr. Math about the simpler case, including general methods, why they work, and applications.

Non-homogeneous Recurrence Relations

(A new question of the week) A recent question asked us to find errors in solving recurrence relations by the method of undetermined coefficients. We’ll see several things that can go wrong, and correct some misunderstandings.

Advanced Logarithm Problems

(A new question of the week) Here is an interesting collection of problems involving logarithms with different bases, which require some unique thinking. And after we’d worked out a good strategy, another problem arose at a whole new level.

Degenerate Conics I: Mystery of the Missing Case

Degenerate cases are instances of a concept that are just on the edge of fitting its definition. They occur when we stretch a definition to its limits, at which point some of the original properties remain, but others break. We’ll start here with common instances of the phenomenon, in conic sections, pursuing the elusive case …

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A Composite Quartic Equation

A question just after we recently discussed quartic equations, has special features that lead to a unique solution method. We’ll be showing how to use synthetic division, and seeing some interesting graphs.