Sequences and Series

[monkeywantsabanana]

Hey guys i'll got a few questions with this topic

First:

If i were given a sequence for say

1, 2, 4, 8

How would i approach this as to finding the possible rule for in terns of and the difference equation? Any thoughts?

I'm still trying to get my head around this. It seems so unnatural to me.

Any help would be appreciated.

[luken93]

Well firstly it's geometric, as it doesn't go up by a constant amount

To find the difference, use

So, using one of any of the combinations,

Also, you know that the first term is (n = 1) is 1

Thus, the sequence is



To check it's right:







etc...

[monkeywantsabanana]

Geometric series?

I have to towards the chapter.

I'm using the Essential book - 5A.

Why do they ask us the question BEFORE they introduce the concept? :O


Is there a specific type for this one?

-1, 8, -27, 64 ?

And thanks luken. (Y)

[luken93]

hahaha they like to do that for that chapter if I remember...

Anyway, you have the two options, geometric series and arithmetic series. Arithmetic goes up/down by a constant difference for each term, but geometric goes up according to ^(n-1), hence the difference isn't arithmetic, but rather geometric...
After you do a few questions, it'll get much easier and a lot more understandable.

For that question, what do you see.

Firstly, they change signs every second term. A possible way to make this occur is if you do the following : (-1)^n, as every second term is positive, which is the same as -1 x -1 = +1

Also, looking at the numbers, you can see that they are cubics, (-1)^3, (2)^-3, (-3)^3 etc

So we can now set up the equation:





To check it's correct:







As I said, it'll come with time, but you will soon be able to spot the trends very quickly..

[monkeywantsabanana]

Haha thanks a lot !

That was much more helpful than my teacher.

Cheers.

[monkeywantsabanana]

Oh hey can you also explain the "difference equation" for the arithmetic and geometric series please?

'cus what you just taught me was only the rule for  in terms of right?

[monkeywantsabanana]

For example this one:



I managed to find out the rule for in terms of (yay!)

but I'm not sure how to find the difference equation.

Help? 

[luken93]

The difference equation is just r

As I showed before, it's the tn / t(n-1), but from memory I can't really remember how to apply it when it's like your example above, although you can clearly see its (1/n)^2...

[bblovee]

if you apply tn/t(n-1) for a few successive terms, you see that each next term is a certain fraction of the previous term
eg for 1, 1/4, 1/9, 1/16:

t2/t1 = 1/4
t3/t2 = 4/9
t3/t4 = 9/16
t5/t4 = 16/25
starting to see a pattern here?
The 5th term is t5 = 1/25, and it is 16/25 of a fraction of the previous term.
Now to express t5 in terms of t4, we have 1/25 = (4^2 / 5^2) * 1/16
so t5 = (4^2/5^2 ) * (t4-1)

you can see that tn = ((n-1)^2/n^2) * (tn-1)
or tn = ( (n-1)/n)^2 * (tn-1)