How to solve this sequence $165,195,255,285,345,x$ This is a question appeared in a competitive exam. The question is:   

Find the unknown term in $165,195,255,285,345,x$   
1)375 $\ \ \ \ \ \ \ \ $               2)420
  3)435 $\ \ \ \ \ \  \ $            4)390   

My Research Effort 
$$a_n =
\begin{cases}
a_{n-1}+30,  & \text{if $n$ is even} \\
a_{n-1}+60, & \text{if $n$ is odd}  \\
\end{cases}$$
where $n>1$
$a_1=165$. In this way the answer should be $a_6=375$. BUT this is not the correct answer. Any hints will be appreciated.
Thanks in advance.
 A: 375 is the answer. Though I don't treat such questions as mathematical question. But they do appear in competitive exams and are base on kind of pattern recognition.
Look at the sequence:
165,(+30)=195 ,(+60)=225,(+30)=285,(+60)=345,(+30)=375
A: I'm saying that $435$ is the answer to the question. Why?
 Consider the polynomial $$ p(x)=-\frac{3x^5}{2}+\frac{55x^4}{2}-\frac{375x^3}{2}+\frac{1175x^2}{2}-786x+525$$
Here is a table of values of $p(x)$ on consecutive $x$.
$\begin{array}{|l|c|c|c|c|c|c|}\hline
x & 1 & 2& 3& 4& 5&6\\ \hline
p(x)&165&195&255&285&345&\color{red}{435}\\\hline
\end{array}
$

My friend is saying that $390$ is the answer. Why? Consider the polynomial
$$g(x)=-\frac{15x^5}{8}+\frac{265x^4}{8}-\frac{1755x^3}{8}+\frac{5375x^2}{8}-\frac{3555x}{4}+570$$
Here is a table of values of $g(x)$ for $x=1,2,3,4,5,6$
$\begin{array}{|l|c|c|c|c|c|c|}\hline
x & 1 & 2& 3& 4& 5&6\\ \hline
g(x)&165&195&255&285&345&\color{red}{390}\\\hline
\end{array}
$

How did I calculate the polynomials?
We are calculating a polynomial $p(x)$ that attains values $165,195,255,285,345,k$ (here $k$ is any number number) when $x=1,2,3,4,5,6$.
I used a principle known as Lagrange Interpolation and the tool Wolframalpha interpolation calculator. 
Similarly one can construct even more complex relations using various interpolation techniques.
Conclusion: There is no unique "next term of the sequence", since for arbitrary number $\lambda$, you can always form a relation in which $\lambda$ should be the next term, although some relations may look more natural than others. 
A: In this case:
consider sequence 
$$
a_n = 15 \cdot p_n,
$$
where $p_n$ is $n$-th prime number.
$a_n$: $\color{gray}{30, 45, 75, 105,} 165, 195, 255, 285, 345, \color{red}{435}, ...$  
A: Another approach but it's just mere an intuitive one.

