<s>[INST] Need some help finding the equation (nth term) of a sequence The sequence goes like this 
Sequence =   1 , 4 , 14 , 58 , 292
Each number is being multiplied by (n+1) and then 2 is added to them. 
Should i proceed to find the equation from 
$ 2(n+1)^n $ 
this? . Thank you very much in advance!
 [/INST] An exponential isn't quite right here, since you are multiplying by a different amount each step. At the first step, $n+1=2$, while at the second step, $n+1=3$, etc. It seems like factorials will be a much better fit for this sequence, since they have the same sort of "multiply by an increasing sequence of numbers behavior. Let's just play around a bit with the first few terms:
$$
1 = 1
$$
$$
4 = 2(1) + 2
$$
$$
14 = 3(2(1) + 2) + 2
$$
$$
58 = 4(3(2(1) + 2) + 2) + 2
$$
Let's take 58 as out example, and expand it out.
We get: $58 = 1(4\cdot3\cdot2) + 2(4\cdot3) + 2(4) + 2$
If we take a bigger example, like 292, we get: $292 = 1(5\cdot4\cdot3\cdot2) + 2(5\cdot4\cdot3) + 2(5\cdot4) + 2(5) + 2$. This expression is just the same as the previous one, but we've multiplied all the terms by 5, and then added another 2 onto the end. I'm noticing that all of the terms here begin with a 2, except for the first one. So let's just rewrite things so the first term also starts with a 2, to make it a bit neater.
$$
292 = 2(5\cdot4\cdot3\cdot2) + 2(5\cdot4\cdot3) + 2(5\cdot4) + 2(5) + 2 - 5!
$$
Here, I've both added and subtracted $5!$, and combined the positive one into the first term. Now we have: $
292 = 2X - 5!
$ where $X = (5\cdot4\cdot3\cdot2) + (5\cdot4\cdot3) + (5\cdot4) + (5) + 1$
Now, if we can just manage to find $X$, we'll be done. Let's rewrite $X$ as the following:
$$
X = \frac{5!}{1!} + \frac{5!}{2!} + \frac{5!}{3!} + \frac{5!}{4!} + \frac{5!}{5!}
$$
I'm noticing the Taylor series for $e$ in that series of fractions. The series for $e$ is $e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \frac{1}{6!} + \frac{1}{7!} + ...$.
Therefore: $5!(e-1) = X + \frac{5!}{6!} + \frac{5!}{7!} + ...$
Notice that that the terms $\frac{5!}{6!} + \frac{5!}{7!} + ...$ are equal to $\frac{1}{6} + \frac{1}{42} + ...$. In other words, their sum is less than $1$. So we could legitimately write the following expression for $X$:
$$
X = \left\lfloor{5!(e-1)}\right\rfloor
$$
Now we did all this for the specific example of the $5^{th}$ term, 292, but we can also make a general formula out of this by replacing the 5s with $n$s.
For $X$, we have: $X = \left\lfloor{n!(e-1)}\right\rfloor$.
So my proposed formula is:
$$
t_n = 2\left\lfloor{n!(e-1)}\right\rfloor - n!
$$
Obviously you still need to prove that this works, and taking the floor to find $X$ might fail for small $n$, meaning you'll need to add some special cases. But this gives you some idea of how one might go about finding a candidate formula in the first place.
</s>