find the 2007th number in the sequence 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,etc.?

Answer 1

The sequence is related to this sum:
1 + 2 + 3 + 4 + 5 + 6 + ...

At the point where the sum is greater than or equal to 2007, you have your answer:

The formula for a sum of the numbers 1 through n is:

n(n+1)
--------- ≥ 2007
... 2

A quick way to estimate this is to realize the n will be close to sqrt(2 * 2007) ≈ 63.356

62 is too low:
62(63)
--------- = 1953
... 2

63 is just right:
63(64)
--------- = 2016
... 2

So we will be in the series of repeating 63s when we are on the 2007th number.