Finding the general term of a sequence (if there's any)

Question

I would like to know if it's possible to find an expression for the following sequence $\;\{a_n\},\;n=0,\,1,\,2,\,3,\,\dots\;$

$1, 3 , 7, 13, 21...$

Someone would like to explain me what I have to do to solve it? I can see a pattern on them, it adds two more than the previous term added (2,4,6,8...) (like 2^n).

So I would like to know the general term($a_n\,=...$) solution, but mainly I would like to the explanation.

Edit: Solution: $a_n\,=n^2-n+1$

Answer 1

HINT: Compare $a_n$ with $n^2$ for the first few values of $n$.

Added:

$$\begin{array}{rcc} n:&1&2&3&4&5&6&7\\ n^k:&1&4&9&16&25&36&49\\ a_n:&1&3&7&13&21&31&43 \end{array}$$

What’s the difference between the last two rows of that table?

Answer 2

This can also be solved as a recurrence relation.

$$\Delta a_n=a_n-a_{n-1}=2(n-1)$$ Sum from $n=2$ to $n=m$: $$\begin{align}\sum_{n=2}^m a_n-a_{n-1}&=2\sum_{n=2}^m (n-1)\\ a_m-\underbrace{a_1}_{1}&=2\sum_{n=1}^{m-1}n=m(m-1)\\ a_m&=m^2-m+1\\ \text{or}\qquad a_n&=n^2-n+1 \end{align}$$