We notice that the sequence seems to be increasing, so let's
start with the 2nd term and divide each term by the previous term







The whole parts of those decimals are going up 2,3,4,... 

So let's see how close multiplying the first term by 2 is to
the second term 8,  2×3 = 6, and we see that we must add 2 to get 8

Now let's see how close multiplying the second term by 3 is to
the third term 27,  8×3 = 24, and we must add 3 to get 27

Now let's see how close multiplying the third term by 4 is to
the fourth term 112,  27×4 = 108, and we must add 4 to get 112

Hmm!  If that pattern works, if we multiply 112 by 5, we should 
have to add 5 to get the next term.  That would be 

112×5 = 560 and if we add 5 we get 565.

But we have to check to see if we multiply 565 by 6 and add 6,
we'll get 3396.

565×6 = 3390 and sure enough when we add 6, we do indeed get 3396.

So the answer for the missing term is 565.

Edwin