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Find r if 1/r + 1 + r + r^2 + r^3 + ... = 4/r.

 Jul 7, 2020
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Notice that the left hand side is a G.P. with first term = 1/r and common ration = r.

 

The left hand side is an infinite sum which only converges when the variable is in a certain bound. 

The bound is -1 < r < 1. Otherwise it will grow to infinity very quickly.

 

Having that in mind, let's use the formula with the contraints -1 < r < 1.

 

Sum of infinite G.P. = \(\dfrac{a}{1 - r}\), where a is the first term and r is the common ratio.

 

So, 

\(\dfrac{\dfrac1r}{1 - r} = \dfrac4r\\ \dfrac{\dfrac1{r} \cdot\color{blue}r\color{black}}{1 - r} = \dfrac{4\color{blue}r\color{black}}r\\ \dfrac1{1 - r}= 4\\ 1 - r = \dfrac14 \)

 

I believe you can continue from here.

 Jul 7, 2020

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