I have the sequence: ($a_1=-1 ,a_{n+1}=a_n+2n , n\in N$)
I'm trying to find an explicit equation for this sequence. I am not familiar to how to find closed form of a recursive sequence so I tried to write first terms of the sequence:
$$\begin{array}{rcc}
n:&1&2&3&4&5&6\\
a_n:&-1&1&5&11&19&29
\end{array}$$
And form here I guessed there is a relation between $n^2$ and $a_n$. and after trying some equations I got: $$a_n=n^2-n-1$$
But as I said earlier, I don't know how to write explicit equation of a recursive sequence. So how can I find this equation mathematically? and how can I learn some basic technics to convert such a sequence to explicit one?(looking for some resources)
Best Answer
You have for all $k \geq 1$ $$a_{k+1}-a_k = 2k$$
so if you sum these equalities for $k=1, ..., n-1$, you get $$\sum_{k=1}^{n-1} (a_{k+1}-a_k) = \sum_{k=1}^{n-1} 2k$$ i.e. $$a_n-a_1 = n(n-1)$$ i.e. finally because $a_1=-1$, $$a_n=n^2-n-1$$