Suites : Exercice 12 2ème année secondaire

1 $U_5=U_0q^5=5\times \left({\dfrac{1}{2}}\right)^5$ $=\dfrac{5}{32}$

2 $U_8=U_0q^8 \iff U_0=\dfrac{U_8}{q^8}$ $=\dfrac{162}{(\sqrt{3})^8}$ $=\dfrac{162}{3^4}=\dfrac{162}{81}=2$

3 $\left\{{\begin{aligned}&{U_3=24}\\&{U_2+U_4=108}\end{aligned}}\right.$ $\iff \left\{{\begin{aligned}&{U_3=24=qU_2}\\&{U_2+qU_3=108}\end{aligned}}\right.$ $\iff \left\{{\begin{aligned}&{U_3=24\;et\;U_2=\dfrac{24}{q}}\\&{U_2+qU_3=108}\end{aligned}}\right.$ $\iff \left\{{\begin{aligned}&{U_3=24\;et\;U_2=\dfrac{24}{q}}\\&{\dfrac{24}{q}+24q=108}\end{aligned}}\right.$
$\dfrac{24}{q}+24q=108$ $\iff 24q^2-108q+24=0$ $\iff 12\left({2q^2-9q+2}\right)=0$ $\iff 2q^2-9q+2=0$
$\Delta=(-9)^2-4\times2\times2=65$ donc $q=\dfrac{9+\sqrt{65}}{4}$ ou $q=\dfrac{9-\sqrt{65}}{4}$