Question

determine a PG crescente de três termos tal que a soma dos 3 termos e 14 e o produto deles è 64​

Answer 1

$\mathrm{Progress\tilde{a}o\ geom\acute{e}trica}\Longrightarrow \left\{\dfrac{x}{q},x,xq\right\}.$

$\left(\dfrac{x}{q}\right)(x)(xq)=64\Longrightarrow x^3=64\Longrightarrow x=\sqrt[3]{64}\ \therefore\ \boxed{x=4}$

$\dfrac{x}{q}+x+xq=14\Longrightarrow x\bigg(\dfrac{1}{q}+1+q\bigg)=14\Longrightarrow 4\left(1+q+q^2\right)=14q$

$\Longrightarrow 2+2q+2q^2=7q\Longrightarrow 2q^2-5q+2=0$

$q=\dfrac{-(-5)\pm\sqrt{(-5)^2-4(2)(2)}}{2(2)}=\dfrac{5\pm3}{4}\Longrightarrow q=\dfrac{1}{2}\ \text{ou}\ q=2$

$\mathrm{A\ PG\ \acute{e}\ crescente}\Longrightarrow q>1\ \therefore\ \boxed{q=2}$

$\left\{\dfrac{x}{q},x,xq\right\}\Longrightarrow\boxed{\left\{2,4,8\right\}}$