Where the heck does the 1 come from?

Hi Smit,

I'm assuming you're referring to the line with (1/7)^x (49 - 1)?

If so, the 1 here is coming from the (1/7)^x being factored out from the previous step. If you re-multiply this expression, you should get the same as the previous line. Without the one, multiplying would result in 49 * (1/7)^x, without the - (1 / 7)^x part.

Here's another example that's not as complex:

5a - a

= 5*a - 1*a

= a * (5 - 1)

I hope this helps! Please let me know if you have any additional questions!

Evan,

Thanks for your help. I'm sorry I'm so slow at this. It's still not clear to me where the 1 is coming from. Shouldn't (1/7)^x-(1/7)^x equal 0? You're subtracting something from itself.

Also, why it this done? If I can understand the why it helps me understand the how.

No, in this case you're not subtracting (1/7)^x from itself because there is the 49 in front of the first (1/7)^x term. In other words, we have 49 of the (1/7)^x terms, and we're subtracting one of those. This would result in 48 of the (1/7)^x terms, which is what you get when you evaluate (49 - 1) in the next step.

Whenever you have something of the form ab - ac, you can factor out the "a" to get "a(b - c)", or written in the other order "(b - c)a". This is read as "a times the result of b minus c". You can think of this as, if we have b a's, and we subtract c of them, then we're left with "b - c" a's. The order that we multiply doesn't matter since multiplication is commutative, e.g. 2 x 3 = 3 x 2.

Does that help?

I think so, yes.

Thanks for your help.

Where does the 1 in (a+1) come from? I'm sorry, I just don't understand.

the distributive property is the underlying principle in this problem. 
I think we should stick to numbers instead of using variables like "a"

for ex: 10(1+3) =10(1)+10(3)

it's important to note that 10(1+3) means 10*(1+3) which is 10 times 1 plus 3. so you do what is in the parenthesis first, 1+3 and then you multiply that by 10 and you get 40, or according to the distributive property you distribute the 10 to the 1 and to the 3, multiplying 10 by 1 and 10 by 3 and then adding the 10 and 30.

So in your original example distributing the (1/7)^x to the 49 and -1 we get (1/7)^x(49)-(1/7)^x(1) which is what the line before says. the key is to remember that (1/7)^x times 1 is just itself. 
Hopefully this helps, if you still don't understand let me know which part of the explanation you don't understand and I can adjust it. 

I'm sorry, I don't know what "zero-based index ordering" is. I've started just adding a rote "1" to the equation, but I still don't understand how it gets there. I hate doing things by rote, but I still (after at least five explanations) don't see how the "1" arrives. 

It's not you, it's me.