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

Feb 1, 2017

The nth term of the sequence is $- 2 \cdot {\left(- 5\right)}^{n - 1}$.

#### 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

$- 2 \cdot {\left(- 5\right)}^{n - 1}$.

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 ${\left(- 5\right)}^{n - 1}$.

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).