(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.
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 problems that take longer to explain: a variation on the rabbit story, and an amazing reverse puzzle.
(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.
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 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.
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 …
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 …