Q. 6 - Calculus

Question

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

Step-by-Step Solution

Answer

The recursive formula is given by

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

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

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

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

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