Before I try and explain: If I ever get confusing I'm going to drop little 'don't forget to ask' bits in. I'm not trying to sound condescending, but it's a pet peeve of mine when a teacher doesn't find it important to clarify a vital but simple point.
Okay well for starts, obviously you get the first step and that's good. Just breaking up the fraction into two parts. You're going to want to try and break those up whenever you can.
The second step was re-ordered/re-indexed. What I mean by that is that the objective of that step was to get all of the terms in both series to the same power of n. The reason they do that is so you can apply the formula for a basic geometric series rather than working out some complex one. (ask if you don't understand, I can clarify)
So after the first step, the first series became
[SM] 4/(4^n)
Lets work with just that one first. For a basic geometric series, the formula is
But that formula will only work provided you have a geometric series in the form of
[SM] r^n
As you can see with the series we're working with, the whole thing is not to the nth power, just the bottom. That is again
[SM] 4/(4^n)
We want to work with [SM]4/(4^n) and manipulate it to make it some power of n. There's two main methods to manipulating a series in order to put it into a form applicable for a formula. That is either to re-order terms, or to draw out terms, which basically would include reordering as well (I can explain more of that as well if you want).
So here, what they did was take the summation and drop the power from n to n-1. They pulled out a term from 4/(4^n).
Imagine 4/(4^n) as 4*(1/(4^n)). It might help to write this out on paper (the algebra) to see it better. You'll be able to see that what was listed in this step, 4*(1/(4^n)) = 4*((1/4)^n).
For some specific reason, the worker of this problem shot for n-1 rather than just n. I guess that makes the formula following slightly less messy, idk. But basically what you've got to see is that 4*((1/4)^n) can be algebraically manipulated even further. You don't necessarily have to though... but it makes it a lot easier on you sometimes.
(1/4)^n-1 =?
(1/4)^n * (1/4)-1 =?
(1/4)^n * 4
They've simply manipulated it. The reason why they've done this is because you want that
[SM] r^n
form. That way, you can apply
For the series and get a quick answer to it (and generally the only way to solve it).
I can go on with the other parts but I think it'd be better if you digested that bit first, especially because it's the same method for the second series. It's simple but I'm very poor with words and I just gave you a very wordy explanation. If you want me to slow down or further explain anything in specific I can. When I find time, I can even draw some of it for you and show you step by step what's going on.
The first step I get, separating the two fractures. Then how do they go from the fractures(not sure if that's the right term, but 4/4^n), to the (1/4) stuff? Is there a rule or what that I missed?
Fuckin' teacher decided it would be funny to add a square root of n under the fracture. And not knowing how to solve that.
I CAN'T HANDLE THIS ANYMORE!!!
I am excausted and I think that today is going to be the first time that I am going to enjoy sleeping!!!
(I love using those faces,even though I should write this one in the "Things I love" thread,sorry)
Thank you very much for the advice,and yes,it is very anoyiong when teachers do that!!!
