<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] As $a_n-a_{n-1}=2n, a_n$ can be at most quadratic.
Let $a_n=An^2+Bn+C$ where $A,B,C$ are arbitrary constants
So, $2n=a_n-a_{n-1}=A(2n-1)+B=2An+B-A$
Comparing the coefficients of $n,2A=2, A=1$
Comparing the constants $B-A=0\implies B=A=1$
So, $a_n= n^2+n +C$
$2=a_1=2+C\implies C=0\implies a_n= n^2+n $
So, $$\sum_{1\le r\le n}a_r= \sum_{1\le r\le n}r^2+ \sum_{1\le r\le n}r= \frac{n(n+1)(2n+1)}6+ \frac{n(n+1)}2=\frac{n(n+1)(n+2)}3$$
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