Question

Give a recursive definition of ${\mathbf{S}}_{\mathbf{m}\left(n\right)}$, the sum of the integer m and the non negative integer n.

Short answer

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

Step-by-step solution

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 Sm (n)

Consider $\mathrm{Sm}\left(\mathrm{n}\right)$is the sum of integer m and the nonnegative integer n.

Then, the equation is:

$S_m\left(n\right)=m+n$

Now, $S_m(n-1)$is the sum of integer m and the nonnegative integer n - 1

such that:

$\begin{array}{r}{S}_m(n-1)=m+(n-1)=m+n-1\\ S_m\left(n\right)=m+n=m+n-1+1=S_m(n-1)+1\end{array}$

Next step is to determine the first term.

The first nonnegative integer is 0.

Therefore, the recursive definition is $S_m\left(0\right)=m,S_m\left(n\right)=S_m(n-1)+1$ for $\mathrm{n}\ge 1$.