Question

Give a recursive definition of ${\mathbf{P}}_{\mathbf{m}}\left(n\right)$, the product of the integer m and the nonnegative integer n.

Short answer

The recursive definition${\mathrm{P}}_{\mathrm{m}}\left(0\right)=0,{\mathrm{P}}_{\mathrm{m}}\left(\mathrm{n}\right)={\mathrm{P}}_{\mathrm{m}}(\mathrm{n}-1)+\mathrm{m}\text{for}\mathrm{n}\ge 1$

Step-by-step solution

The recursive definition of the sequence: Step 1: Define recursive sequence

A sequence can also be defined recursively, meaning that the previous terms define successive terms in the sequence. The recursive sequence is obtained by the deriving each successive term such that it is 2 larger than the previous obtained term.

To find the recursive definition of Pm (n)

Consider $P_m\left(n\right)$is the product of integer m and the nonnegative integer n as

$P_m\left(n\right)=mn$.

Now, $P_m(n-1)$is the product of integer m and the nonnegative integer n - 1 . Then, the equation is:

Then, the equation is:

$$\begin{gathered} {\mathrm{P}}_{\mathrm{m}}(\mathrm{n}-1)=\mathrm{m}(\mathrm{n}-1)=\mathrm{mn}-\mathrm{m} \\ \mathrm{Also},{\mathrm{P}}_{\mathrm{m}}\left(\mathrm{n}\right)=\mathrm{mn}=\mathrm{mn}-\mathrm{m}+\mathrm{m}={\mathrm{P}}_{\mathrm{m}}(\mathrm{n}-1)+\mathrm{m} \end{gathered}$$

Consider the first term non-negative integer as: $P_m\left(0\right)=m\cdot 0=0$

Therefore, by the recursive definition${\mathrm{P}}_{\mathrm{m}}\left(0\right)=0,{\mathrm{P}}_{\mathrm{m}}\left(\mathrm{n}\right)={\mathrm{P}}_{\mathrm{m}}(\mathrm{n}-1)+\mathrm{m}\mathrm{for}\mathrm{n}\ge 1$