Question

Find a closed form for the generating function for the sequence\(\left\{ {{a_n}} \right\}\), where,

a) \({a_n} = 5\) for all\(n = 0,1,2, \ldots \).

b) \({a_n} = {3^n}\)for all\(n = 0,1,2, \ldots \)

c) \({a_n} = 2\)for\(n = 3,4,5, \ldots \)and\({a_0} = {a_1} = {a_2} = 0\).

d) \({a_n} = 2n + 3\)for all\(n = 0,1,2, \ldots \)

e) \({a_n} = \left( {\begin{array}{*{20}{l}}8\\n\end{array}} \right)\)for all\(n = 0,1,2, \ldots \)

f) \({a_n} = \left( {\begin{array}{*{20}{c}}{n + 4}\\n\end{array}} \right)\)for all\(n = 0,1,2, \ldots \)

Short answer

(a) The required result is\(\frac{5}{{1 - x}}\)

(b) The required result is\(\frac{1}{{1 - 3x}}\).

(c) The required result is\(\frac{{2{x^3}}}{{1 - x}}\).

(d) The required result is\(\frac{{3 - x}}{{{{(1 - x)}^2}}}\)

(e) The required result is\({(1 + 2x)^7}\).

(f) The required result is\(\frac{1}{{{{(1 - x)}^5}}}\).

Step-by-step solution

Formula of generating function

Generating function for the sequence \({a_0},{a_1}, \ldots ,{a_k}\)of real numbers is the infinite series;

\(G(x) = {a_0} + {a_1}x + {a_2}{x^2} + \ldots + {a_k}{x^k} + \ldots = \sum\limits_{k = 0}^{ + \infty } {{a_k}} {x^k}\)

Use the definition of a generating function and solve the sequence

For the sequence:

\({a_n} = 5\)for all\(n = 0,1,2, \ldots \).

\(\begin{array}{l}{a_n} = 5\\n = 0,1,2, \ldots \end{array}\)

Use the formula for generating function:

\(\begin{array}{l}G(x) = 5 + 5x + 5{x^2} + \ldots \\G(x) = \frac{5}{{1 - x}}\end{array}\)

Use the definition of a generating function and solve the sequence

For the sequence:

\({a_n} = {3^n}\)for all \(n = 0,1,2, \ldots \)

Use the formula for generating function:

\(\begin{array}{l}G(x) = 1 + 3x + 9{x^2} + 27{x^3} + 81{x^4} + 243{x^5} + 729{x^6} + \ldots \\G(x) = \sum\limits_{k = 0}^{ + \infty } {{{(3x)}^k}} \\G(x) = \frac{1}{{1 - 3x}}\end{array}\)

Use the definition of a generating function and solve the sequence

For the sequence:

\({a_n} = 2\)for \(n = 3,4,5, \ldots \) and \({a_0} = {a_1} = {a_2} = 0\)

Use the formula for generating function:

\(\begin{array}{l}G(x) = 0 + 0x + 0{x^2} + 2{x^3} + 2{x^4} + \ldots \\G(x) = 2{x^3} \cdot \frac{1}{{1 - x}}\\G(x) = \frac{{2{x^3}}}{{1 - x}}\end{array}\)

Use the definition of a generating function and solve the sequence

For the sequence:

\({a_n} = 2n + 3\)for all \(n = 0,1,2, \ldots \)

Use the formula for generating function:

\(\begin{array}{l}G(x) = 3 + 5x + 7{x^2} + \ldots \\G(x) = \frac{2}{{{{(1 - x)}^2}}} + \frac{{1 - x}}{{{{(1 - x)}^2}}}\\G(x) = \frac{{3 - x}}{{{{(1 - x)}^2}}}\end{array}\)

Use the definition of a generating function and solve the sequence

For the sequence:

\({a_n} = \left( {\begin{array}{*{20}{l}}8\\n\end{array}} \right)\)for all \(n = 0,1,2, \ldots \)

Use the formula for generating function:

\(G(x) = \left( {\begin{array}{*{20}{l}}8\\0\end{array}} \right) + \left( {\begin{array}{*{20}{l}}8\\1\end{array}} \right)x + \left( {\begin{array}{*{20}{l}}8\\2\end{array}} \right){x^2} + \ldots + \left( {\begin{array}{*{20}{l}}8\\8\end{array}} \right){x^8} + 0{x^9} + 0{x^{10}} + \ldots \)

\(G(x) = \sum\limits_{k = 0}^8 {\left( {\begin{array}{*{20}{l}}8\\k\end{array}} \right)} {x^k}\)

\(G(x) = {(1 + x)^8}\)

Use the definition of a generating function and solve the sequence

For the sequence:

\({a_n} = \left( {\begin{array}{*{20}{c}}{n + 4}\\n\end{array}} \right)\)for all \(n = 0,1,2, \ldots \)

Use the formula for generating function:

\(\begin{array}{c}G(x) = \sum\limits_{k = 0}^{ + \infty } {\left( {\begin{array}{*{20}{c}}{k + 4}\\k\end{array}} \right)} {x^k}\\G(x) = \sum\limits_{k = 0}^{ + \infty } {\left( {\begin{array}{*{20}{c}}{5 + k - 1}\\k\end{array}} \right)} {x^k}\\G(x) = \frac{1}{{{{(1 - x)}^5}}}\end{array}\)