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Going over the homework, make sure to mention #3(e), an alternating series . You get that kind of alternation by throwing in a $(-1{)}^{n}$ or, in this case, $(-1{)}^{n-1}$ .
Last night’s homework ended with the series “all the even numbers between 50 and 100.” Some students may have written $\sum _{n=1}^{\text{26}}(\text{48}+\mathrm{2n})$ . Others may have written the answer differently. But one thing they probably all agree on is that adding it up would be a pain. If only there were...a shortcut!
Let’s consider the series $3+5+7+9+11+13+15+17$ . (Write that on the board.)
What do we get if we add the first term to the last? Answer: 20. Modify your drawing on the board to look like this:
OK, what about the second term to the second-to-last? Hmm....20 again. Add to the drawing, and then keep adding until it looks like this:
So, looking at that drawing, what does $3+5+7+9+11+13+15+17$ add up to? Hopefully everyone can see that it adds up to four 20s, or 80.
And this is the big one—will that trick work for all series? If so, why? If not, which series will it work for? Answer: It will work for all arithmetic series . The reason that the second pair added up the same as the first pair was that we went up by two on the left, and down by two on the right. As long as you go up by the same as you go down, the sum will stay the same—and this is just what happens for arithmetic series.
OK, what about geometric series? Write the following on the board:
$2+6+18+54+162+486+1458$
Clearly the “arithmetic series trick” will not work here: $2+1458$ is not $6+486$ . We need a whole new trick. Here it comes. First, to the left of your equation, write $S=$ so the board looks like:
$S=2+6+18+54+162+486+1458$
where $S$ is the mystery sum we’re looking for. Now, above that, write:
$3S=$
ask the class what comes next. Can we just multiply each term by 3? (Yes, distributive property.) When you write this line, line up the numbers like this:
$3S=6+18+54+162+486+1458+437$
$S=2+6+18+54+162+486+1458$
But don’t go too fast on that step—make sure they see why, if $S$ is what we said, then $3S$ must be that!
Now, underline the second equation (as I did above), and then subtract the two equations . What do we get on the left of the equal sign? What do we get on the right? See how things cancel? See if you can get the class to tell you that...
$2S=4374-2$
So then $S$ is just 2186. They may want to verify this one on their calculators. Once again, however, the key is to understand why this trick always works for any Geometric series.
“Homework: Arithmetic and Geometric Series”
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