confusion about recurrence relation let us suppose that  we have following recurrence relation 
$a_n-a_{n-1}=2*n$  where  $a_0=0$ 
as i know solution is following
$a_n=a_0+\sum\limits_{i=1}^n 2n   $ 
which is the same  as
$a_n=2*\sum\limits_{i=1}^n n   $ 
but this term
$ \sum\limits_{i=1}^n n=n^2$
therefore 
$a_n=2*n^2$ 
but youtube says that 

and also  wolfram alpha says that solution is  $a_n=n*(n+1)$  how?

The following two summations are correct,
\begin{align}
\sum_{i=1}^n i &= \frac{n(n+1)}{2} \\
\sum_{i=1}^n n &= n \sum_{i=1}^n 1 = n(n) = n^2
\end{align}
For your recurrence relation, we have that at each iteration you extract a term of the form $2i$ as such
\begin{equation}
\begin{split}
a_n &= a_{n-1} + 2n\\ &= a_{n-2} + 2(n-1) + 2n\\ &= \vdots \\ &= a_{i} + 2(i+1) + \ldots + 2n
\end{split}
\end{equation}
For $i = 0$, we get
$$a_n = a_0 + \sum_{i=1}^n 2i = 0 + 2 \sum_{i=1}^n i = 2 \frac{n(n+1)}{2}=n(n+1)$$
