What number comes next in the sequence $7, 16, 8, 27, 9,...$? What number comes next in this series?
$$7, 16, 8, 27, 9,...$$
I thought it was $38$, but I'm wrong.  
It is a multiple choice, and options are $27, 10, 40, 37$.  
Don't worry - I'm not cheating on anything, but helping my daughter with homework.
And I can't help her since I can't figure it out myself!
 A: To me, it's $40$.
Indeed
$$16 = 2\cdot 8$$
$$27 = 3\cdot 9$$
thence
$$40 = 4\cdot 10$$
The even terms of the series may follow this path, so a possible series could be
$$7, 16, 8, 27, 9, 40, 10, 55, 11, 72, 12, \cdots$$
A: One possibility is that the $1$st, $3$rd, $5$th, and so on, numbers are just increasing by one, so 
$$7,\_,8,\_,9,\_,10,\_,...$$
Now notice that $16=2 \cdot 8$, and $27=3 \cdot 9$. Therefore it would be reasonable to assume the next number to be $40=4 \cdot 10$.
A: The next term is again $27$; then we had a palindromic sequence
$$
7,16,8,27,9,27,8,16,7,\ldots,
$$
repeating like this. 
A: $40$, because the sequence is $n+7$ and $n^2+10n+16$ interleaved.
$37$, because the sequence is $n+7$ and $-n^2/2+23n/2+16$ interleaved.
Use imagination and you come with equally good excuses for the other two options.
A: Another view, also giving $40$ as the next term: $2n+2;\frac{n}{2};3n+3;\frac{n}{3};4n+4;\frac{n}{4}\cdots$
\begin{align}\text{initial value}&=7\\
7*2+2&=16\\
\frac{16}{2}&=8\\
8*3+3&=27\\
\frac{27}{3}&=9\\
9*4+4&=40\\
\frac{40}{4}&=10
\end{align}
Or: $2(n+1);\frac{n}{2};3(n+1);\frac{n}{3};4(n+1);\frac{n}{4}\cdots$
\begin{align}\text{initial value}&=7\\
2(7+1)&=16\\
\frac{16}{2}&=8\\
3(8+1)&=27\\
\frac{27}{3}&=9\\
4(9+1)&=40\\
\frac{40}{4}&=10
\end{align}
