Question

Find the generating function for the finite sequence$\mathbf{2}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{2}$ .

Short answer

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

Step-by-step solution

Formula of generating function

For a finite sequence , 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;

$\mathit{G}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{C}\mathbf{(}\mathit{n}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{+}\mathit{C}\mathbf{(}\mathit{n}\mathbf{,}\mathbf{1}\mathbf{)}\mathit{x}\mathbf{+}\mathit{C}\mathbf{(}\mathit{n}\mathbf{,}\mathbf{2}\mathbf{)}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{\cdots }\mathbf{+}\mathit{C}\mathbf{(}\mathit{n}\mathbf{,}\mathit{n}\mathbf{)}{\mathit{x}}^{\mathbf{n}}$

By the binomial theorem, this is $\mathbf{(}\mathbf{1}\mathbf{+}\mathit{x}{\mathbf{)}}^{\mathbf{n}}$.

Define the generating function

The generating functions forgiven by:

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

This can be rewritten as:

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