<s>[INST] Next number in series What are the basic/advanced strategies used to find the next number in series.
I know the simple ones such as addition, multiplication etc. But recently I came into a question that goes on something like 812, 819, 823, 835, 834, 851(Don't try to solve this, I changed some numbers and there is no sequence). This is to illustrate that the sequence goes up and goes down at 835 then goes back up. What strategies can you use to solve it? 
These questions are for an aptitude test so you have about 60-90 secs to solve each question.
P.S. I can't really post the question that was asked as I might be disqualified from the job process if I post it. 
 [/INST] About your series:
Let 
$$
\begin{eqnarray*}
\begin{split}
P(x)&:= 812\frac{(x-2)(x-3)x-4)(x-5)(x-6)}{(1-2)(1-3)(1-4)(1-5)(1-6)}+819\frac{(x-1)(x-3)(x-4)(x-5)(x-6)}{(2-1)(2-3)(2-4)(2-5)(2-6)}\\
&+823\frac{(x-1)(x-2)(x-4)(x-5)(x-6)}{(3-1)(3-2)(3-4)(3-5)(3-6)} +835\frac{(x-1)(x-2)(x-3)(x-5)(x-6)}{(4-1)(4-2)(4-3)(4-5)(4-6)}\\
&+834\frac{(x-1)(x-2)(x-3)(x-4)(x-6)}{(5-1)(5-2)(5-3)(5-4)(5-6)}+851\frac{(x-1)(x-2)(x-3)(x-4)(x-5)}{(6-1)(6-2)(6-3)(6-4)(6-5)} 
\end{split}
\end{eqnarray*}
$$
Then $P(x)$ is a polynomial which fits your data. This method is called Lagrange interpolation method.
BUT if $f(x)$ is any function defined on the integers, then 
$$Q(x)=P(x)+(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)f(x)$$
also fits your data.
Most of these problems are actually asking for the simplest formula which fits that data, but really is there such a mathematical thing as "simplest". Lets say that our sequence is $2,3,5$ ( I picked only 3 to keep things simple). Which one would you say that it is the simplest description? 


*

*$2^{n-1}+1$

*$\frac{n^2-n+4}{2}$

*the nth prime..

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