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# help

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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