Question

Give a recursive definition for K! for integers k ≥ 0. Be sure you define 0! as part of your answer.

Short answer

The recursive formula is given by

$k!=\left\{\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}k=0\\ \prod _{i=1}^k i\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}k>0\end{array}\right.$

Step-by-step solution

Step1. Given Information

$\text{Definition for}k!\text{:}$

$\text{If}k\text{is a non-negative integer then the product of first}k\text{integers is defined as}$

$k!=1\cdot 2\cdot 3\cdot 4\cdots k$.

$\text{For}k=0\text{, defining}0!=1\text{.}$

Step2. Recursive formula

$\text{Formulate recursive relation for}k!\text{.}$

$\text{The recursive formula is given by}$

localid="1650619256590" role="math" $k!=\left\{\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}k=0\\ \prod _{i=1}^k i\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}k>0\end{array}\right.$