Magic and Arithmetic Series

Now your high schoolers have learned the two formulas for the sum of an AP. (Do they still call it an Arithmetic Progression?)  So you ask them this:

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Arithmetic sequences and series. I cannot think of a more mind-numbing introduction to them than the way it was done a century ago when I was in school. And a quick googling suggests that the situation may not have improved very much — what I see often begins with a caveat that “You won’t actually need this until you take Calculus.”  Hard on the heels of this are the definitions of the first and last terms, the common difference, and so on. Then comes the formula for the general term, and finally the iconic derivation of the formula for sum of an arithmetic progression.  

Like many math teachers, I also used to tell my students the story of the clever young Gauss. It probably firmed up their belief that you have to be born with a math brain in order to do math.  Raise a glass to Kate Nowak for what she did to introduce AP's. In fact it is her post that prodded me to write this.

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If I were able to tardis back a few years, I would probably begin with this: 

Give me the sum of three consecutive numbers.

If the students were to tell me that the sum is 72, I would tell them immediately that the three numbers are 23, 24, and 25. 

And if then I might offer this:

Give me the sum of three consecutive even numbers.

If the sum is 84, I would tell them that the three numbers are 26, 28, and 30.

Perhaps even this:

Give me the sum of five consecutive numbers.

For example, if the sum is 45, I would tell them immediately that the five numbers are 7, 8, 9, 10, and 11.

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The secret to this is that whenever you have an arithmetic series with an odd number, n, of terms, the sum is always n times the middle term.

It is easy to convince kids that this is the case for the simple cases given above. For example, three consecutive numbers with a middle term m must always be of the form

m - 1,     m ,      m + 1.

Adding, we get 3m. To perform the trick, divide the sum by 3, and subtract 1 to get the first number.

For three consecutive even numbers, the situation is pretty much the same. The three numbers would be 

m - 2,     m,     m + 2,

Adding, the sum is again 3m.

In case you are wondering about doing the trick when you are given the sum of four consecutive numbers: dividing that sum by 2 gives the sum of the middle pair of numbers from which you can easily deduce what the four numbers are.

For example, if the sum is 50, then the sum of the middle pair is 25, so the middle pair is 12 and 13, from which we get the four numbers 11, 12, 13, 14. 

The sum of an arithmetic series with an even number, n, of terms is always n/2 times the sum of the middle pair. Interestingly, the sum of the middle pair is also the sum of the first term and last term. 

You can pursue this far enough to derive the two formulas for the sum of an arithmetic progression, but I’m not sure that I would push it that far.

Instead I would switch the question around to finding the sum of a longer list of consecutive numbers, à la Kate Nowak.

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The possibility of introducing arithmetic sequences and series in this manner grew out of a simple trick from William Simon’s book Mathematical Magic.  

A calendar trick

Ask a student to draw a rectangle around three consecutive dates on a calendar month, like so:

This is to be done so that you, the teacher, cannot see what dates have been encircled (for example, have the student stand behind your back while you are facing the rest of the class). Ask the volunteer to show the circled dates to the rest of the students (but not to you) and to tell you the sum of the dates. You can immediately announce the dates that have been circled. 

[A personal aside: I am notoriously poor at mental arithmetic — a brief description of my troubles is contained here. Using a calendar appeals to me because it forces the three numbers to be small, thus avoiding the floundering that would occur if some cheeky person asks me “What are the three numbers if the sum is 14691?”] 

After explaining how to do the three-in-a-row trick,  the calendar itself might prompt students to ask questions like this:

How would you do it if we gave you the sum of four consecutive dates?

How would you do it if we gave you the sum of three dates in a vertical line? 

What if we gave you the sum of five consecutive dates? 

The second and third questions the students could answer themselves.

When there are four consecutive dates, there is no middle date and the sum of the four dates is not divisible by 4. I imagine this might be a stumbling block for some students. But, fingers crossed, at least some of them will actually look at what happens when they do divide by 4, and thereby open up other avenues to explore. 

They might, for example, note that dividing by 4 gives the number that is the average of the middle pair of dates.  For the four dates circled above, when I divide the sum by 4, I get 5.5, which is the average of the two middle dates 5 and 6.  And not only that, 5.5 is also the average of the first and last dates.

That is: 

Which generalizes to 

But as I said earlier, I probably wouldn't push it this far.

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What is the sum of the following arithmetic progression? Each square represents a term in the progression. 

What about this one? 

Or this one?