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

1 Answer
Feb 1, 2017

Answer:

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).