## A crack at Dynamic Programming

Dynamic Programming is a discrete optimization technique. It means, the variables we want to calculate using this method are discrete. As an example, if x is one such variable then $x\in\{1,2,3,4\}$ is acceptable, but ‘x is a real number between 0 and 1′ is NOT acceptable since there are infinitely many real numbers between 0 and 1. In math terms, we can say that the domain of the variable x is a countable set.

Problems that are solved by dynamic programming are recursive nature. Let’s look at the problem that ask us to calculate the sum up to a number, i.e., if $f_s$ is such a function then $f_s(5)=0+1+2+3+4+5$. The recursive definition (assuming i is non negative) is following:

$f_{s}(i)=\begin{cases}0 & ,i=0\\i+f_{s}(i-1) & ,i\geq1\end{cases}$