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This is what I would do:

• Count how many integers (N)
• Choose 0 as the next number
• Create a N-order polynomial that goes through all the integers
• Claim that this is the pattern of the series

So for your example of

2 4 8 ?

The answer is

2 4 8 0

Which is clearly a logical relation, since it follows this equation perfectly:

(-7/3.)*i**3 + 8*i**2 + (-11/3.)*i + 2

Where i is in the index in the sequence. I found this with python/numpy:

x = array([0, 1, 2, 3])
y = array([2, 4, 8, 0])
polyfit(x, y, deg=3)

They might say this is not a real relation and ask for more numbers, so you might want to play it safe and use N+10 with all zeros. So:

  -0.000700 * i**9 +
   0.030010 * i**8 +
  -0.545073 * i**7 +
   5.452083 * i**6 +
 -32.632755 * i**5 +
 118.526042 * i**4 +
-251.163536 * i**3 +
 275.991865 * i**2 +
-113.657937 * i    +
   2.000000

2 4 8 0 0 0 0 0 0 0

Advantages of this approach:

• It works for all sequences
• It's easy to automate
• The sequence is easy to remember, just add zeros
• You probably won't get the job, so big time savings right there
• Everyone will tell you that you must be fun at parties