Question

1- Escreva uma PG de três termos, sabendo que a sua soma é – 3/2 e o produto entre eles vale 1.

Answer 1

$\boxed{\begin{array}{l}\sf P~G\bigg(\dfrac{x} {q},x,xq\bigg)\\\\\sf\dfrac{x}{\red{\backslash\! \! \! \!} q}\cdot x\cdot x\cdot\red{\backslash\! \! \! \!} q=1\\\\\sf x^3=1\\\sf x=\sqrt[\sf3]{\sf1}\\\sf x=1\\\sf\dfrac{1}{q} +1+q=-\dfrac{3}{2}\cdot(2q)\\\sf 2+2q+2q^2=-3q\\\sf 2q^2+2q+3q+2=0\end{array}}$

$\boxed{\begin{array}{l}\sf 2q^2+5q+2=0\\\sf\Delta=b^2-4ac\\\sf\Delta=5^2-4\cdot2\cdot2\\\sf\Delta=25-16\\\sf\Delta=9\\\sf q=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\sf q=\dfrac{-5\pm\sqrt{9}}{2\cdot2}\\\\\sf q=\dfrac{-5\pm3}{4}\begin{cases}\sf q_1=\dfrac{-5+3}{4}=-\dfrac{2}{4}=-\dfrac{1}{2}\\\\\sf q_2=\dfrac{-5-3}{4}=-\dfrac{8}{4}=-2\end{cases} \end{array}}$

$\boxed{\begin{array}{l}\sf para~q=-\dfrac{1}{2}:~\bigg(-2,1,-\dfrac{1}{2}\bigg)\\\\\sf para~q=-2:~\bigg(-\dfrac{1}{2},1,-2\bigg) \end{array}}$