Math Help - Geometric Series 1 of 2

1. Geometric Series 1 of 2

The sum to infinity of a geometric progression is 5 and the sum to infinity of another series formed by taking the first, fourth, seventh and tenth,.....terms ( that is U1 + U4 + U7 +U10 +......) is 4. Find the common ratio of the first series. Answer: r = (-1+sq Root 2) / 2 Why is r = (-1-sq root 2)/2 rejected as an answer!

2. Hello, puggie!

The sum to infinity of a geometric progression is 5, and the sum to infinity
of another series formed by taking the 1st, 4th, 7th, 10th, ... is 4.

Find the common ratio of the first series.

Answer: . $r \:=\:\frac{\text{-}1 + \sqrt{2}}{2}$

Why is $r = \frac{\text{-}1-\sqrt{2}}{2}$ rejected as an answer? . . To converge, ${\color{red}|r| < 1}$

The first series has first term $a$ and common ratio $r.$
. . Its sum is: . $S_1 \:=\:\frac{a}{1-r} \:=\:5 \quad\Rightarrow\quad a \:=\:5(1-r)\;{\color{blue}[1]}$

The second series has first term $a$ and common ratio $r^3.$
. . Its sum is: . $\frac{a}{1-r^3} \:=\:4 \quad\Rightarrow\quad a \:=\:4(1-r^3)\;{\color{blue}[2]}$

Equate ${\color{blue}[1]}$ and ${\color{blue}[2]}$: . $5(1-r) \:=\:4(1-r^3) \quad\Rightarrow\quad 4r^3 - 5r + 1 \:=\:0$

. . which factors: . $(r-1)(4r^2 + 4r-1) \:=\:0$

. . and has roots: . $r \;=\;1,\;\frac{\text{-}1+\sqrt{2}}{2},\;\frac{\text{-}1-\sqrt{2}}{2}$

And only $r \:=\:\frac{\text{-}1 +\sqrt{2}}{2}$ qualifies as a common ratio for a convergent series.