Help with Proof and Mathematical Induction problem(adsbygoogle = window.adsbygoogle || []).push({});

Here is my problem I need to solve:

"Prove that the statement: [tex] \frac {1}{5} + \frac{1}{5^2} + \frac{1}{5^3} +... + \frac{1}{5^n} = \frac{1}{4}(1-\frac{1}{5^n})[/tex] is true for all positive integers [tex]n[/tex]. Write your proof in the space below."

I don’t know where to start. The sub-chapter in my textbook that talks about it doesn’t make any sense. Here are the two pages relevant to this problem.

My first question is why is [tex]n[/tex] redefined as [tex]k[/tex]? It seems pointless. Why not just say “show the statement is true for the next integer [tex]n+1[/tex]”?

Second question: When [tex]n=1[/tex] you just replace a 1 for all the spots where an [tex] n [/tex] appears in the equation; so why on earth when you make it [tex]n=k+1[/tex] don’t you just put in a 2 wherever an [tex]n[/tex] is (or [tex]k[/tex]) instead of actually inserting the whole [tex]k+1[/tex]? It seems like they make it more complicated then it needs to be.

Third question: What the heck is with the [tex] x + x + x +[/tex] … [tex]+ x= [/tex] part? I don’t understand what it is supposed to mean or how it is relevant to the problem given in the example (or in my problem I need to answer).

Fourth question: why, in example 1, do they add a [tex] (k+1)^2[/tex] to each side of the problem? Isn’t the whole point to just solve the equation with more than one integer to prove that it is true for all integers? I don’t get it!

Fifth question: I don’t understand the conclusion of example 1. They just say “This proves that… for all positive integers [tex]n[/tex].” How does it prove it? They didn’t even use real numbers? It is just a bunch of gibberish!

Thoroughly confused.

Maybe if I can understand the example that they give me I can have a fighting chance at answering this exam question (which I need to show my work for).

Thanks,

Alan

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# Help with Proof and Mathematical Induction problem

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