<< Chapter < Page Chapter >> Page >
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:

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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: Arithmetic and Geometric Series”

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Advanced algebra ii: teacher's guide. OpenStax CNX. Aug 13, 2009 Download for free at http://cnx.org/content/col10687/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Advanced algebra ii: teacher's guide' conversation and receive update notifications?