# Number sequence 6

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.

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:
$$172 + 80/30 = 172 + 8/3 = 174\frac{2}{3}$$

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.