$\begin{aligned}\begin{cases}u_1.u_5=25\\ u_2+u_3+u_4=31\end{cases}&\Leftrightarrow \begin{cases}u_1.u_1.q^4=25\\ u_1.q+u_1.q^2+u_1.q^3=31\end{cases}\\ &\Leftrightarrow \begin{cases}\left(u_1.q^2\right)^2=25\\ u_1.q+u_1.q^2+u_1.q^3=31\end{cases}\\\ &\Leftrightarrow \begin{cases}u_1.q^2=\pm5\\ u_1.q+u_1.q^2+u_1.q^3=31\end{cases}\end{aligned}$

Trường hợp 1: $u_1.q^2=5$
$\begin{aligned}\begin{cases}u_1.q^2=5\\ u_1.q+5+u_1.q^3=31\end{cases}&\Leftrightarrow\begin{cases}u_1.q^2=5\\ u_1.q+u_1.q^3=26\end{cases}\\ &\Leftrightarrow \begin{cases}u_1.q^2=5 &(1)\\ u_1.q\left( 1+q^2\right)=26 &(2)\end{cases}\end{aligned}$
Chia (2) cho (1) ta được
$$\begin{aligned}\dfrac{u_1.q\left(1+q^2\right)}{u_1.q^2}=\dfrac{26}{5}&\Leftrightarrow\dfrac{1+q^2}{q}=\dfrac{26}{5}\\ &\Leftrightarrow5q^2-26q+5=0\\
&\Leftrightarrow\left[\begin{array}{l}
q=5 \\ q=\dfrac{1}{5}\end{array}\right.\\ &\overset{(1)}{\mathop{\Leftrightarrow }}\left[ \begin{array}{l}u_1=\dfrac{5}{q^2}=\dfrac{1}{5} \\ u_1=\dfrac{5}{q^2}=125\end{array}\right.\end{aligned}$$

Trường hợp 2: $u_1.q^2=-5$
$\begin{aligned}\begin{cases}u_1.q^2=-5\\ u_1.q-5+u_1.q^3=31\end{cases}&\Leftrightarrow\begin{cases}u_1.q^2=-5\\ u_1.q+u_1.q^3=36\end{cases}\\ &\Leftrightarrow\begin{cases}u_1.q^2=-5 &(3)\\ u_1.q\left(1+q^2\right)=36 &(4)\end{cases}\end{aligned}$
Chia (4) cho (3) ta được:
$$\begin{aligned}\dfrac{u_1.q\left(1+q^2\right)}{u_1.q^2}=-\dfrac{36}{5}&\Leftrightarrow\dfrac{1+q^2}{q}=-\dfrac{36}{5}\\ &\Leftrightarrow 5q^2+36q+5=0\\ &\Leftrightarrow\left[\begin{array}{l}q=\dfrac{-18+\sqrt{299}}{5}\\ q=\dfrac{-18-\sqrt{299}}{5}\end{array}\right.\\ &\overset{(3)}{\mathop{\Leftrightarrow}}\left[\begin{array}{l}u_1=-\dfrac{5}{q^2}=\dfrac{-125}{623-36\sqrt{299}}\\ u_1=-\dfrac{5}{q^2}=\dfrac{-125}{623+36\sqrt{299}}\end{array}\right.\end{aligned}$$