Question

Calcule a razão de uma P.G. cujo 2º termo é igual a 12 e a soma do 1º com o 3º é – 51

Com calculo por favor!

Answer 1

Olá Letícia,

$\begin{cases}a_2=12\\ a_1+a_{3}=-51\end{cases} \begin{cases}a_1\cdot q=12\\ a_1+(a_1\cdot q^2)=-51 \end{cases}\begin{cases}a_1= \dfrac{12}{q}~~(I)\\ a_1+(a_1\cdot q^2)=-51~~(II) \end{cases}\\\\ Substitua~I~em~II:\\\\ \dfrac{12}{q}+\left( \dfrac{12}{q}\cdot q^2\right)=-51\\\\\\ \dfrac{12}{q}+ \dfrac{12q^2}{q}=-51\\\\\\ \dfrac{12}{q}+12q=-51\\\\\\ 12+12q^2=-51q\\ 12q^2+51q+12=0$

$\Delta=51^2-4\cdot12\cdot12\\ \Delta=2.601-576\\ \Delta=2.025\\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ q'=\dfrac{-51+45}{24}= \dfrac{-6}{24}= -\dfrac{1}{4} \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nearrow\\ q= \dfrac{-51\pm \sqrt{2.025} }{2\cdot12}= \dfrac{-51\pm45}{24}\\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\searrow~\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~q''= \dfrac{-51-45}{24}= \dfrac{-96}{~~24}=-4$

Portanto esta P.G. possui duas razões,  $- \dfrac{1}{4}~~e~~-4$.

Tenha ótimos estudos ;D