Closed form generating function

Question

Q. What is the generating function for the sequence 1,1,1,1,1,1? Ans. The generating function for the sequence is $1+x+x^2+x^3+x^4+x^5.$

Now we have ** $\frac{(x^6-1)}{(x-1)} = 1+x+x^2+x^3+x^4+x^5$.**

Consequently, $G(x) = \frac{x^6-1}{x-1}$ is the required generating function.

I don't understand the line closed by **. What method has been applied to get that.

Answer 1

for any $f(x) = 1+x+x^2+x^3+\cdots+x^n$

Multiply both sides by by $(x-1)$

You'll have $f(x)(x-1) = x^{n+1}-1$

$=> f(x) = \frac{x^{n+1}-1}{x-1}$

Answer 2

See,

$\frac{x^6-1}{x-1}= \frac{(x-1)(x^5+x^4+x^3+x^2+x+1)}{x-1}=x^5+x^4+x^3+x^2+x+1$

Answer 3

Every polynomial with one indeterminate whose coefficients are all $1$, that is, $$x^n+x^{n-1}+\cdots+1$$ is the quotient $$\frac{x^{n+1}-1}{x-1}$$

You can just make the division using Ruffini's rule. Or you can consider the polynimial as a sum of a geometric progression and apply the formula.