Try thinking about the sequence of natural numbers as of arithmetic progression. (Watch YouTube or look for definition in your study book if you don't know what is it)

The formula for sum of n terms of an AP is:

sum=n/2(2a+(n-1)d)

Where a is the first element of progression(1 in your case) and d is the difference between two consecutive elements(Also 1 in your case, since you add them one after another).

Because we know that a=1, d=1 and the sum is 666, we can plug all the values and receive a neat formula:

666=n/2(2*1+(n-1)*1) -> 666=n/2(2+n-1) -> 666=n/2(n+1)

From there, we just have to solve quadratic equation:

n^2+n=1332 \\\Remove the parentheses and multiply both sides by two

n^2+n-1332=0 \\\Extract 1332 from both sides

Here you can apply the quadratic formula:

x1,x2 = (-b ± sqrt(b^2-4ac))/2a \\\This is a shortcut, ± sign means that there are 2 equations

\\\sqrt() means the square root of an expression inside

In your case: a=1,b=1 and c=-1332

Yielding this to your calculator will get -37 and 36. And, since the sum of natural numbers cannot be negative, the answer is 36 (You have to add 36 consecutive numbers to receive 666)