@The Mask:

The method you used was essentially the method I used. Only reason it took me over forty seconds to see it was because I got caught up in 6*9=42, which is correct, but only in base 13 math. So I lost time that way.

I feel like a math geek surrounded by guys who are even more geeky at math than me. I mean, square this and square that? And I'm stuck with simply finding out a pattern for the numbers. Never mind algebra and differential equations; all I've got is finding a darn pattern with simple numbers.

You guys are just amazing.

3263442

Square the number and then add itself to the result, repeat. Took somewhere between 30 seconds and a minute to figure it out.

3,263,442.

Time: Around 5-10 minutes. I tend to do these things on paper to analyze it.

Method:
Add one to the given number. The sum will then become the multiplier while the original number will become the multiplicand. The product will be the next number in the set.
1
1 + 1=2 ---> 1 x 2=2
2 + 1=3 ---> 2 x 3=6
6 + 1=7 ---> 6 x 7=42
42 + 1=43 ---> 42 x 43 = 1806
1806 + 1 = 1807 ---> 1806 x 1807 = 3,263,442

I didn't get it until after I peeked at people's answers. Whoops.

I'm tired and don't like working too hard when I'm at home which is where I relax which doing math does not equal. So there.

3263442

Time: Under 30 seconds, and above 10 (didn't time or anything, lol).

Two divisions and it's done! You see that 6 x 7 is 42, and 42 x 43 is 1806.

I love math. Math, math, math.

*huggles his Differential Equations textbook*

Hahahahahhaa...

@Allamorph:

Whatever I think of first is what happens, and it tends to be more complicated. Screw algebra.

@Katana:

Incidentally, wouldn't it be easier to look at it as x(x+1) instead of x^2+x? One less mechanical step, since you can do the addition mentally.

3263442

Method: Square the number, add itself to get the next number. IE, 6^2 + 6 = 42. 42^2 + 42 = 1806 and so on.

Time: Twenty seconds to figure it out. Much longer to punch it in on the computer's calculator to check my theory.