Number sequence 6

Question

Fill in the correct number in this sequence:

$16,32,40,88,92,172,?$

The options are: $178$,$176$,$174\frac{2}{3}$,$177$, and $175\frac{1}{3}$

At first, I thought the pattern was $2x+8$ but that does not work out.

Source: a publicly available practice test in a book.

Answer 1

Take the differences $A(n+1) - A(n)$. This gives the following sequence:
$16, 8, 48, 4, 80$
We can note that:
$16/2 = 8, 8*6 = 48, 48/12 = 4, 4*20 = 80$, i.e.
$a(n)/2 = a(n+1), a(n+1)*6 = a(n+2), a(n+2)/12 = a(n+3), a(n+3)*20 = a(n+4)$
So we get alternating division and multiplication using a 3rd sequence:
$2, 6, 12, 20$ which looks like this oeis sequence. ($0, 2, 6, 12, 20, 30$).
So the next number is 30. Now going in reverse order we have:
$a(n+4)/30 = a(n+5)$, or $80/30 = a(n+5)$. So in the original sequence the next number will be:
$172 + 80/30 = 172 + 8/3 = 174\frac{2}{3}$

Answer 2

How about this?

Count the holes in the decimal digits
16 -> 1
32 -> 0
40 -> 2
88 -> 4
92 -> 1
172 -> 0

1, 0, 2, 4, 1, 0, X
X=2 (by simple repetition)

So, ? has 2 holes. The only option that fits the bill is 178.

...

If a 4 has zero holes, then we have 1, 0, 1, 4, 1, 0, X and then 176 would be the only option instead.