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Formulas

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.

A Tunnel Through the Earth

I have a very short problem this week: How deep will you go if you dig a straight tunnel through the earth, how long will it be, and what angle do you have to start at?

Summing Squares: Finding or Proving a Formula

Last week we looked at problems about counting the squares of all sizes in a checkerboard. Some solutions required finding the sum of consecutive squares, 1^2+2^2+3^2+\dots+n^2, for which we used a formula whose derivation I deferred to this week. Here we’ll see a couple proofs that require knowing the formula ahead of time, and a …

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Arithmetic Series, Backward

Here is a recent question about arithmetic sequences and series (specifically, reversing the process to find the number of terms given the sum), that nicely illustrates a common type of interaction with a student: gathering information about both problem and student, then guiding them to use what they know, or giving new information as needed. …

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Distances on Earth 3: Planar Approximation

We’ve looked at two formulas for the distance between points given their latitude and longitude; here we’ll examine one more formula, which is valid only for small distances. This is a “flat-earth approximation” to distance.

Cutting Up Space Using n Planes

As the capstone of this series on counting, lets look at something that’s a little harder to count by drawing: What is the maximum number of regions into which all of 3-dimensional space can be divided by n planes? We’ll look at two significantly different perspectives.