Question 1111116
<pre>
n   a_n      diff     diff_2    diff_3
1   2             
2   12        10
3   36        24       14
4   80        44       20         6   
5   150       70       26         6 


Because the differences don't reach a constant for 3 levels, the form will be

{{{ a[n] = k[3]n^3 + k[2]n^2 + k[1]n + k[0] }}}

Using the first four values, you get these 4 equations:
    {{{ k[3]  + k[2] + k[1] + k[0] = 2 }}}
   {{{ 8k[3]  + 4k[2] + 2k[1] + k[0] = 12 }}}
   {{{ 27k[3]  + 9k[2] + 3k[1] + k[0] = 36 }}}
   {{{ 64k[3]  + 16k[2] + 4k[1] + k[0] = 80 }}}
 
Which results in the following matrix equation:

  {{{ (matrix(4,4, 1,1,1,1, 8,4,2,1, 27,9,3,1,  64,16,4,1)) * (matrix(4, 1, k[3], k[2], k[1], k[0])) = (matrix(4,1, 2,12,36,80))  }}} 



Which can be solved using matrix inversion to find:
{{{ k[3] = 1 }}}
{{{ k[2] = 1 }}}
{{{ k[1] = 0 }}}
{{{ k[0] = 0 }}}

Therefore  {{{ highlight(  a[n] = n^3 + n^2  )}}}

——
Check:
 n=1:   {{{ a[n] = 1^3 + 1^3 = 2 }}}  (ok)
 n=5:  {{{ a[n] = 5^3 + 5^2 = 125 + 25 = 150 }}}  (ok)