Algebra

In general, in a geometric series, three terms, say $a{r}^{k}, a{r}^{k+1}, a{r}^{k+2}$, then $a{r}^{k}*a{r}^{k+2}=({a{r}^{k+1}})^{2}$, or, otherwise, in three terms of a geometric series, the middle term squared, equals the product of the two next to it.

Using this, we get the equation

$(60+x)(140+x)={(120+x)}^{2}$, where x is the added number.

$8400+200x+{x}^{2}=14400+240x+{x}^{2}$. Cancelling out the x²:

$8400+200x=14400+240x$

$40x=-6000$

$x=-150$.

The resulting series will be:

$-90, -30, -10$.

The common ratio is $\frac{-30}{-90}=\frac{1}{3}$. So, the resulting sequence has a common ratio of 1/3.