## Domain and Range of a Radical Function

(A new question of the week) We’ve looked at domain and range problems before, but some have more interesting details than others. Here is a superficially basic radical function (and the answer is extremely easy when you just use a graphing tool), which raised some interesting issues while solving it algebraically.

## Fibonacci Word Problems II: Challenging

Last week we looked at several basic word problems for which the Fibonacci sequence is part of the solution. Now we’ll look at two problem that take longer to explain: a variation on the rabbit story, and an amazing reverse puzzle.

## The Case of the Disappearing Derivative

(A new question of the week) An interesting question we received in mid-January concerned two implicit derivative problems with an unusual feature: the derivative we are seeking disappears! How do you track down such elusive quarry? Each case is a little different.

## Fibonacci Word Problems I: Basic

Here and next week, we’ll look at a collection of word problems we have seen that involve the Fibonacci sequence or its relatives, sometimes on the surface, other times only deep down. The first set (here) are direct representations of Fibonacci, while the second set will be considerably deeper. A 20-foot walkway The first, from …

## Interpreting Probability Questions

(A new question of the week) A couple recent questions centered around how to interpret probability problems, whose wording can often be subtle, and whose solutions require care.

## Generalizing and Summing the Fibonacci Sequence

Continuing our look at the Fibonacci sequence, we’ll extend the idea to “generalized Fibonacci sequences” (with different starting numbers), and see that the ratio of consecutive terms is the same in general as in the usual special case. Then we’ll look at the sum of terms of both the special and general sequence, turning it …

## A Few Inductive Fibonacci Proofs

Having studied proof by induction and met the Fibonacci sequence, it’s time to do a few proofs of facts about the sequence. We’ll see three quite different kinds of facts, and five different proofs, most of them by induction. We’ll also see repeatedly that the statement of the problem may need correction or clarification, so …

## Graph Coloring: Working Through a Proof

(A new question of the week) The Math Doctors have different levels of knowledge in various fields; I myself tend to focus on topics through calculus, which I know best, and leave the higher-level questions to others who are more recently familiar with them. But sometimes, both here and in my tutoring at a community …

## The Golden Ratio and Fibonacci

We’re looking at the Fibonacci sequence, and have seen connections to a number called phi (φ or $$\phi$$), commonly called the Golden Ratio. I want to look at some geometrical connections and other interesting facts about this number before we get back to the Fibonacci numbers themselves and some inductive proofs involving them.

## Introducing the Fibonacci Sequence

We’ve been examining inductive proof in preparation for the Fibonacci sequence, which is a playground for induction. Here we’ll introduce the sequence, and then prove the formula for the nth term using two different methods, using induction in a way we haven’t seen before.