13.3 Arithmetic and geometric series

 Page 1 / 1
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 $\left(-1{\right)}^{n}$ or, in this case, $\left(-1{\right)}^{n-1}$ .

Last night’s homework ended with the series “all the even numbers between 50 and 100.” Some students may have written $\sum _{n=1}^{\text{26}}\left(\text{48}+2n\right)$ . 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:

“Homework: Arithmetic and Geometric Series”

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers!