Deriving the geometric series sum formula

[preet]

I want to understand how the formula for the sum of a geometric sequence is created... This is what I understand so far:

A geometric sequence is the sum of a series of numbers, where a term will be multiplied by an amount (the common ratio) to get the next term, and so on... ex: 1+2+4+8...64+128+256
I understand that the first term is 1 and the common ratio is 2...

The formula to find the sum of the series is

SN=a(1-rN)/(1-r)

Where S is the sum for the 'n'th term...

Step by step, they show the formula worked out like this:

1) a + ar^1 + ar^2 + ar^3 + ar^4 ... ar^n-2 + ar^n-1

2) multiply the whole thing by 'r' ... ar + ar^2 + ar^3 + ar^4 ... ar^n-1 + ar^n

3) subtract the two sequences

4) end up with a - ar^n = (1-r) SN

5) rearrange to get SN=a(1-rN)/(1-r)

Okay, so I don't understand anything from 2 down... if you have a sequence in front of you how can you just think "Why don't I just multiply the whole series by its common ratio and subtract it from the first series to find its sum?" ... what's the reasoning behind multiplying it and then cancelling out most of the terms by subtracting? How do you just do something like that out of the blue?

Thanks in advance,
Preet

 

[Tide]

It might help if you actually wrote those things out as equations. Both sides of the equation represent numbers, namely the sum that you're interested in. You basically have S = stuff.

It should be no surprise that if two numbers are equal to each other then multiplying both of them by the same quantity will yield an equation that is just as valid as the first.

So you have Sum = stuff and Q X Sum = Q X stuff. Now if you subtract the equations from each other (left side from left side and right side from right side) then the resulting equation will be true because you're subtracting the same quantity from the same number!

Your equation follows - and NO it is not out of the clear blue. People basically see the pattern and arrive at the logical way to exploit that pattern. You just need to study it for a while.

 

[preet]

"

Okay, can someone walk me through this with simple numbers than?
I've used

15 = 1+2+4+8 as an example... a=1, r=2

30 = 2+4+8+16

Of course the second expression will be true but why is it just multiplied by the common ratio?

I worked out both of the above series' to -15 = -15 (when I subtracted)... this is where I am right now... (Tide) said that people can arrive at a logical way to exploit the pattern... I'm obviouisly having a lot of trouble with this... so I'd appreciate any help...

 

[Tide]

Did you notice when you subtracted the two equations that ALL the terms canceled except the first and last?

 

[Hurkyl]

Pretend that you're unable to add four numbers together, so that you're unable to directly determine that 1 + 2 + 4 + 8 = 15. Since we cannot determine the value, let's give it a name: S. So, we have

S = 1 + 2 + 4 + 8

What happens when you apply the steps to this?