Re: (Discrete Math - Induction) 'Formula Differentiation'

On Sun, 30 Mar 2008 20:48:11 -0700 (PDT), "almeidabatista@xxxxxxx"
<almeidabatista@xxxxxxxxx> wrote:

On 30 mar, 23:45, quasi <qu...@xxxxxxxx> wrote:
On Sun, 30 Mar 2008 18:30:51 -0700 (PDT), "almeidabati...@xxxxxxx"

<almeidabati...@xxxxxxxxx> wrote:
Hi all! I've got this problem in my set:

'1 + 2q + 3q^2 + ... + nq^(n-1) = [1 - (n+1)q^n + nq^(n-1)]/[(1 -
q)^2], q <> 1.

Estabilish [the formula above] by differentiating the expansion of the
formula for the sum of a geometric progression.'

I've been thinkering with this one all day, no clue on how to start.
ANY hints on how the derivative of the sum of terms of a G.P will get
into this are welcome.

(1) Find an antiderivative of the LHS.

(2) Look at the result -- do you recognize it?

(3) Based on the answer to (2), use a known formula to simplify the
result.

(4) Now differentiate the simplified result,

quasi

Hmmmmmm! This algebric trick would never happen to me wouldn't it be
your answer! Thanks a lot!

Just to be sure, the only 'induction' involved in the solution is
finding the derivative of the summation?

Unless you're required to be ultra-formal, there's no need for
induction. Just show the pattern, making clear that it works, term by
term. If you want, you can also show explicitly what happens to the
k'th term, where k is an arbitrary index variable, left undetermined.

quasi
.

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