Question

Resolva as equações: a) x/5 + 2x/15 +
4x/45 + ...... = 9

                                              b)
x + x/2 + x/4 + ..... = 5

Answer 1

Quando o módulo da razão de uma P.G está entre 0 e 1, podemos ter a soma infinita dos termos de uma P.G:

$S_{n}=\dfrac{a_{1}}{1-q}$
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a)

$a_{1}=x/5\\a_{2}=2x/15\\\\q=a_{2}/a_{1}=(2x/15)/(x/5)=(2x/15)*(5/x)=2/3$

2/3 está entre 0 e 1, logo:

$\dfrac{a_{1}}{1-q}=S_{n}$

Como essa soma dá 9:

$\dfrac{a_{1}}{1-q}=9\\\\\\\dfrac{\frac{x}{5}}{1-\frac{2}{3}}=9\\\\\\\dfrac{\frac{x}{5}}{(\frac{1}{3})}=9\\\\\\\dfrac{x}{5}=9*\dfrac{1}{3}\\\\\\\dfrac{x}{5}=3~~~\therefore~~~x=3*5~~~\therefore~~~\boxed{\boxed{x=15}}$

b)

$a_{1}=x\\a_{2}=x/2\\\\q=a_{2}/a_{1}=(x/2)/x=(x/2)*(1/x)=1/2$

1/2 está entre 0 e 1:

$\dfrac{a_{1}}{1-q}=S_{n}~~~\therefore~~~a_{1}=S_{n}*(1-q)\\\\\\a_{1}=5*(1-q)\\\\x=5*(1-\frac{1}{2})\\\\x=5*(\frac{1}{2})\\\\\boxed{\boxed{x=\frac{5}{2}}}$