The sum 27

The sum of a geometric progression's second and third terms is six times the fourth term. Find the two possible values of the common ratio.

Correct answer:

r₁ =  0.5
r₂ =  -0.3333

Step-by-step explanation:

$$\begin{gathered} a_2+a_3=6a_4 \\ {a}_2+r\cdot a_2=6\cdot r^2\cdot a_2 \\ 1+r=6\cdot r^2 \\ 1+r=6\cdot r^2 \\ -6r^2+r+1=0 \\ 6r^2-r-1=0 \\ a=6;b=-1;c=-1 \\ D=b^2-4ac=1^2-4\cdot 6\cdot (-1)=25 \\ D>0 \\ {r}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{1\pm \sqrt{25}}{12} \\ {r}_{1,2}=\frac{1\pm 5}{12} \\ {r}_{1,2}=0.083333\pm 0.416667 \\ {r}_1=0.5 \\ {r}_2=-0.333333333 \end{gathered}$$

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