How do you write an equation for the nth term of the geometric sequence -2,10,-50,...?

1 Answer
Feb 1, 2017

The nth term of the sequence is -2 * (-5)^(n-1).

Explanation:

Dividing a term with the previous one gives the result -5. This means that to get the next term, we multiply by -5. The sequence also starts at -2, which is quite unconventional to express in terms of -5. Therefore, since the problem doesn't restrict us in any way, we can say that the equation is

-2 * (-5)^(n-1).

This is because, like I mentioned above, to get to the next term we need to multiply by -5. Therefore, to get to the nth term, we need to multiply by (-5)^(n-1).

As for why the exponent is n-1 and not just n, this is because for n= 1, we are talking about the first term, which doesn't have anything to do with -5. Because of this, it's convenient to "cover" the first term with our equation by utilizing the fact that

a^0 = 1 for any real nonzero a.

In other words, in order for the equation to include the case of the first number -2 (when n=1) we need an expression that multiplies -2 with not n, but n-1 instances of -5, otherwise the first term would be 10 (and that's our second term, and so on).