Question

qual o valor de x na na pg x-1, 2x, 4x-1

Answer 1

PG = {x-1 ; 2x ; 4x-1}

A razão de uma PG se encontra dividindo um termo pelo termo anterior. Encontraremos x calculando q (razão).

$q= \dfrac{4x-1}{2x} ~~~~e~~~~q = \dfrac{2x}{x-1} \\\\\\ Iguale~~as~~express\~oes:\\\\\\ \dfrac{4x-1}{2x}=\dfrac{2x}{x-1}\to~~~~multiplique~~~cruzado:\\\\\\ (4x-1)(x-1)=(2x)(2x)\to~~~4x^2-4x-x+1=4x^2\to\\\\ 4x^2-4x^2-5x+1=0\to~~~-5x=-1~(-1)\to~~~5x=1\to\\\\ \large\boxed{x = \dfrac{1}{5} }$


Para conferir, basta substituir o valor em x, e verificar se forma mesmo uma PG.


$a_1=x-1\to~~~ \dfrac{1}{5} -1\to~~~\dfrac{1-5}{5} \to~~~-\dfrac{4}{5} \\\\\\a_2=2x\to~~~2\left(\dfrac{1}{5} \right)\to~~~\dfrac{2}{5} \\\\\\ a_3=4x-1\to~~~ 4\left(\dfrac{1}{5} \right)-1 \to~~~ \dfrac{4}{5} -1 \to~~~ \dfrac{4-5}{5}\to~~~ -\dfrac{1}{5}\\\\\\ \Large\boxed{PG=\left\{ -\dfrac{4}{5}~~;~~ \dfrac{2}{5} ~~;~ -\dfrac{1}{5} \right\} }$


Calculando a razão:


$q = \dfrac{a_3}{a_2} \to~~~ \dfrac{-\dfrac{1}{5} }{\dfrac{2}{5}} \to~~~ -\dfrac{1}{5}\times\dfrac{5}{2}\to~~~ -\dfrac{5}{10}\to~~~\boxed{q=-\dfrac{1}{2}}\\\\\\ q = \dfrac{a_2}{a_1} \to~~~ \dfrac{\dfrac{2}{5} }{-\dfrac{4}{5}} \to~~~ \dfrac{2}{5}\times \left(-\dfrac{5}{4}\right) \to~~~- \dfrac{10}{20}\to~~~\boxed{q=-\dfrac{1}{2}}$




Ok, o valor encontrado para x está correto. Espero ter ajudado =)