in a geometric sequence, the second term is $\frac{-4}{5}$ sum of first three terms :$\frac{38}{25}$ . What is the first term? 
In a geometric sequence, the second term is $\frac{-4}{5}$
    and the sum of the first three terms is $\frac{38}{25}$ . What is the value of the sequence's first term?

for some reason I keep getting a decimal as my answer which I'm pretty sure it can't be because it was a math homework problem:
My Steps
if we call the first term to be the number $n$ and $r$ to be the multiplyer
we have the following:
$n+nr+nr^2=\frac{38}{25}$ 
and then we multiply r on both sides to get the following
$nr+nr^2+nr^3=\frac{38r}{25}$
and then we substitute nr with $\frac{-4}{5}$
and we get the following:
$\frac{-4}{5}+\frac{-4r}{5}+\frac{-4r^2}{5}=\frac{38r}{25}$
but then when solving for the number $r$ I do not get a pretty number and I am sure that I am making some sort of mistake. I was wondering what I was doing wrong?
Also I understand that because this is a quadratic, I was wondering which "answer" to choose from the zero's?
 A: \begin{align}
   -\dfrac 45\left( \dfrac 1r + 1 + r \right) &= \dfrac{38}{25} \\
   \dfrac 1r + 1 + r &= -\dfrac{19}{10}\\
   \dfrac 1r + \dfrac{29}{10} + r &= 0 \\
   10+29r +10r^2 &= 0 \\
   (2r+5)(5r+2) &= 0 \\
   r &\in \left\{-\dfrac 52, -\dfrac 25 \right\}
\end{align}  There are two solutions because the sequence and the sequence reversed will both have the same sum and will both be geometric sequences.
So the first term is either 
$\left(-\dfrac 45\right)\left(-\dfrac 52 \right) = 2$ or 
$\left(-\dfrac 45\right)\left(-\dfrac 25 \right) = \dfrac{8}{25}$
A: Beginning from $$\frac{-4}{5}+\frac{-4r}{5}+\frac{-4r^2}{5}=\frac{38r}{25}$$
we clear denominators by multiplying by 25:
$$-20 + -20r + -20r^2 = 38r$$
$$20r^2+58r+20=0$$
Now divide by $2$:
$$10r^2+29r+10=0$$
This factors as
$$(5r + 2 )(2r + 5) = 0$$
which gives $r=-2/5$  and $r = -5/2$ as two solutions.  Both are valid so there is no reason to choose one over the other.
Once you've got $r$, it should be straightforward to find $n$.
A: Following what you wrote Mr. John:
$$1+r+r^2=-\frac{19}{10}r.$$ Then
$$r^2+2.9r+1=0$$
We can easily find the two roots $r_1 = -2.5$ and $r_2  = -0.4$.
Now, since $nr = -\frac{4}{5}$ then $r_1 = -2.5$ will give  $n = 0.32 = \frac{8}{25}$.
And $r_2 = -0.4$ will give  $n = 2$. 
