The easiest way to explain this is to say: The first term is -2 and each number in the sequence after that is the previous one times -3. That is called a recursive definition and would look like this:
f(1)=-2
f(n)=F(n-1)*-3
Once we start with f(1) we can just plug it into the second line to get f(2)=6, f(3)=-18, and so on. But if we want a single equation that will let us know f(100) without calculating all 99 previous numbers in the sequence, that formula is pretty much useless.
So let's think about this. Looking at the sequence we see that:
f(2)=-2*-3= -2*(-3)^1
f(3)=-2*-3*-3=-2*(-3)^2
f(4)=-2*-3*-3*-3=-2*(-3)^3
Notice that the exponent that -3 is getting raised to is 1 less than the sequence number we want. It gets raised to 1 when n is 2, and so on. That also works in our favor as x^0=1 so that f(1)=-2*(-3)^0.
That leads us to the final equation:
f(n)=-2*(-3)^(n-1)
