Consider the sequence $a_1 = 2, a_2 = 5, a_3 = 9, a_4 = 14,$ etc…
(a) The recurrence relation is: $a_1 = 2$ and $a_n = a_{n – 1} + (n + 1) \; \forall \;n \in [\mathbb{Z \geq 2}]$.
(b) Conjecture an explicit formula for $a_n$. (Proof for conjecture pretermitted here)
I wrote out some $a_n$ to compass to cotton on to an idea or pattern. It seems bootless.
$\begin{array}{cc}
a_2 = 5 & a_3 = 9 & a_4 = 14 & a_5 = 20 & a_6 = 27\\
\hline \\
5 = 2 + (2 + 1) & 9 = 5 + (3 + 1) & 14 = 9 + (4 + 1) & 20 = 14 + (5 + 1) & 27 = 20 + (7 + 1) \\
\end{array}$
The snippy answer only says $a_n = (n^2 + 3n)/2$. Thus, could someone please explicate the (missing) motivation or steps towards this conjecture? How and why would one envision this?
Best Answer
Write out the series for $a_{n}$ to start with. We have that
$a_{n} = a_{n-1} + (n+1)\\ \quad = a_{n-2} + n + (n+1) \\ \quad = \ldots \\ \quad = a_{1} + 3 + 4 + \ldots + (n+1) \\ \quad = 2 + 3 + 4 + \ldots + (n+1) \\ \quad = \displaystyle \left(\sum_{i=1}^{n+1} i \right) - 1 \\ \quad = (n+1)(n+2)/2 - 1, \quad (\text{using the value for the sum of the integers between 1 and}\ n + 1) \\ \quad = (n^{2} + 3n)/2 + 1 -1 \\ \quad = (n^{2} + 3n)/2$