<s>[INST] Summation of sequence $a_n - a_{n-1} = 2n$ $(a_1,a_2,a_3,..)$ be a sequence such that $a_1$ =2 and $a_n - a_{n-1} = 2n$ for $n \geq 2$. Then $a_1 + a_2 + .. + a_{20}$ is equal to?
$a_1$ = 2 
$a_2$ = 2 + 2x2
$a_3$ = 6 + 2x3
$a_4$ = 12 + 2x4
$a_5$ = 20 + 2x5
$a_n$ = $b_n$ + $g_n$ ,here $g_n$=2xn 
$b_3$-$b_2$=4 
$b_4$-$b_3$=6
$b_5$-$b_4$=8
$b_6$-$b_5$=10
 [/INST] I have 2 methods of doing this:
Method one:
$a_2-a_1=4$, i.e. $a_2=6$
$a_3-a_2=6$, i.e. $a_3=12$
$a_4-a_3=8$, i.e. $a_4=20$
You will notice that $a_k = 2+4+6+8+ \cdots +2k = \frac{k}{2}(2k+2)=k^2 + k$ (using summation formula)
$a_1+a_2+ \cdots + a_{20} = \displaystyle\sum_{k=1}^{20} (k^2+k)=\frac{(20)(20+1)(2\times20+1)}{6}+20\times\frac{1+20}{2}=3080$
</s>