How to get the result of only first N terms of a geometric series like $1+Ax+Ax^2+Ax^3+Ax^4+Ax^5$....

Question

Given -1 < x < 1, and for Series like the following, I am trying to figure out not the complete Total, but only the Total of first N Terms.

So the Question is:

What is the Total of first N Terms in:$$1 + Ax + Ax^2 + Ax^3 + Ax^4 + Ax^5 + Ax^6...$$

and

What is the Total of first N Terms in:$$1 - Ax + Ax^2 - Ax^3 + Ax^4 - Ax^5 + Ax^6...$$

Thank You

Answer 1

You have to factor $Ax$:

$$1+Ax(1+x+x^2+x^3+\cdots+x^{N-2})$$

the parenthesis is equal to $\frac{x^{N-1}-1}{x-1}$

So you got:

$$1+A\dfrac{x^{N}-x}{x-1}$$

For the second sum, you have to separate the "Positive $+$" terms and the "Negative $-$" and each individual sum is like the first

Answer 2

In general, the sum of the first $n$ terms of a geometric series is:

$$ s = a + ar + ar^{2} + \ldots + ar^{n-1} = a \left( \frac{1-r^{n}}{1-r} \right) $$

provided that $r \neq 1$.

In your first question, take $r = X$.

In your second question, take $r = -X$.

Then solve.