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A teacher's guide to lecturing on arithmetic and geometric series.

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 n = 1 26 ( 48 + 2n ) size 12{ Sum cSub { size 8{n=1} } cSup { size 8{"26"} } { \( "48"+2n \) } } {} . 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:

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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.

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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.

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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:

3 S =

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:

3 S = 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 3 S 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...

2 S = 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:

“Homework: Arithmetic and Geometric Series”

Questions & Answers

what is the stm
Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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LITNING Reply
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LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Rafiq
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Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
Daniel
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
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Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Source:  OpenStax, Advanced algebra ii: teacher's guide. OpenStax CNX. Aug 13, 2009 Download for free at http://cnx.org/content/col10687/1.3
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