Expert: Paul Klarreich - 1/13/2011

Question
Hey Paul,

I really don't know where to begin.

Let r be a real number, r ≠ 1. Prove that for every integer n ≥ 1,

1+ r+ r^2+r^3+...+r^(n-1) = (r^n-1)/(r-1).

Would I turn the left side into 1+(r^(n-1))!  and solve from there, or should I use the well ordered axiom and principles of complete induction? If so, how would include a P(0) to prove this equation on be true?

Thanks for your time.

Answer
Questioner: Ashley
Country: United States
Category: Advanced Math
Private: No
Subject: Algebra
Question: Hey Paul,

I really don't know where to begin.

Let r be a real number, r ≠ 1. Prove that for every integer n ≥ 1,

1 + r+ r^2+r^3+...+r^(n-1) = (r^n-1)/(r-1).

Would I turn the left side into 1+(r^(n-1))!

>>> NO, NO, NO.  It does not say that.  Factorial is a product, not a sum.

and solve from there, or should I use the well ordered axiom and principles of complete induction? If so, how would include a P(0) to prove this equation on be true?

Thanks for your time.
....................................
This is your standard 'sum of a geometric series'.  Try this:

Let

A  = 1 + r + r^2 + r^3 + ... + r^(n-1)

then

rA =     r + r^2 + r^3 + ... + r^(n-1) + r^n

Now subtract:

rA - A =  r + r^2 + r^3 + ... + r^(n-1) + r^n
       -(1 + r + r^2 + r^3 + ... + r^(n-1))


      =      r + r^2 + r^3 + ... + r^(n-1) + r^n
       - 1 - r - r^2 - r^3 + ... - r^(n-1)

= r^n - 1

And if:

rA - A = r^n - 1

you can solve for A:

A(r - 1) = r^n - 1

and finish it up.
........................................
Induction works, too, of course.  

P(0) is the statement:
    r^(0+1) - 1
1  = ------------
     r - 1
That left side is the series that begins with 1 and ends with r^0, which is also 1, so it has only one term.

Do it.  If you get stuck, let me know.

Write P(k)
Prove P(k+1), using P(k)

but you knew that, didn't you?