Question

Find the generating function for the finite sequence 2,2,2,2,2.

Short answer

The required result is $2\left(\frac{x^6-1}{x-1}\right)$.

Step-by-step solution

Formula of generating function

For a finite sequence${\mathit{a}}_{\mathbf{0}}\mathbf{,}{\mathit{a}}_{\mathbf{1}}\mathbf{,}\mathbf{\dots }\mathbf{,}{\mathit{a}}_{\mathbf{k}}$, the generating sequence is;

$\mathit{G}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{a}}_{\mathbf{0}}\mathbf{+}{\mathit{a}}_{\mathbf{1}}\mathit{x}\mathbf{+}{\mathit{a}}_{\mathbf{2}}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{\cdots }\mathbf{+}{\mathit{a}}_{\mathbf{k}}{\mathit{x}}^{\mathbf{k}}$

For the sequence ${\mathit{a}}_{\mathbf{k}}\mathbf{=}\mathit{C}\mathbf{(}\mathit{n}\mathbf{,}\mathit{k}\mathbf{)}$ for $\mathbf{0}\mathbf{⩽}\mathit{k}\mathbf{⩽}\mathit{n}$, the generating function is;

G(x) = C(n,0) + C(n,1)x + C(n,2)x²+ .... + C(n,n)xⁿ

By the binomial theorem, this is (1 + x)ⁿ .

Define the generating function

The generating functions for$2,2,2,2,2$given by:

$2+2x+2x^2+2x^3+2x^4+2x^5$

This can be rewritten as:

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