Q:

Find the sum to $n$ terms of the series $10+84+734+....$

Find the sum to n terms of the series $10+84+734+....$

$\frac{9(9^n+1)}{10} + 1$
$\frac{9(9^n-1)}{8} + 1  $
$\frac{9(9^n-1)}{8} + n $
$ \frac{9(9^n-1)}{8} + n^2$

My attempt:
I'm getting option $(4)$.
For $n=1$, all options are right,
For $n=2$, sum must be $10+84=94$.

Can you explain in formal way? Please.

A:

The terms are neither geometric nor arithmetic progression. There could be more than one recurrence relations that can describe the $n^{th}$ term, but it is unlikely that you could deduce it based on $3$ terms only. Since this is only a multiple choice questoon, I suspect that they expect you to find out the right choice using your method.
On the mathematical side, we can find out the general term $T(n)$ by subtracting the sum of first $n-1$ terms from the sum of first $n$ terms. (By the way I suppose the option is $\frac{9(9^n-1)}8+n^2$ 
$$T(n)=\frac{9^{n+1}-9}8 + n^2  - \frac{9^n-9}8 -(n-1)^2$$
$$T(n)=9^n +2n -1$$
Which is pretty hard to guess based on the first three terms only

