The sum 21

The sum of a geometric progression's 2nd and 3rd terms is six times the 4th term. Find the two possible values of the common ratio B. If the second term is eight, the common ratio is positive. Find the first six terms.

Correct answer:

q₁ =  0.6404
q₂ =  -0.3904
g₁ =  12.4922
g₂ =  8
g₃ =  5.1232
g₄ =  3.2809
g₅ =  2.1011
g₆ =  1.3455

Step-by-step explanation:

$$\begin{gathered} g_2+g_3=4g_4 \\ {g}_2+q\cdot g_2=4\cdot q^2\cdot g_2 \\ 1+q=4\cdot q^2 \\ 1+q=4\cdot q^2 \\ -4q^2+q+1=0 \\ 4q^2-q-1=0 \\ a=4;b=-1;c=-1 \\ D=b^2-4ac=1^2-4\cdot 4\cdot (-1)=17 \\ D>0 \\ {q}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{1\pm \sqrt{17}}{8} \\ {q}_{1,2}=0.125\pm 0.515388 \\ {q}_1=0.640388203=0.6404 \\ {q}_2=-0.390388203 \end{gathered}$$

Our quadratic equation calculator calculates it.

$$\begin{gathered} g_2=8 \\ q>0 \\ q=q_1=0.6404 \\ {g}_1=g_2/q=8/0.6404=12.4922 \end{gathered}$$

$g_3=q\cdot g_2=0.6404\cdot 8=5.1232$

$g_4=q\cdot g_3=0.6404\cdot 5.1232=3.2809$

$g_5=q\cdot g_4=0.6404\cdot 3.2809=2.1011$

$$\begin{gathered} g_6=q\cdot g_5=0.6404\cdot 2.1011\doteq 1.3455 \\ \text{ Verifying Solution: } \\ {t}_2=g_2+g_3-4\cdot g_4=8+5.1232-4\cdot 3.2809\doteq -0.0004 \end{gathered}$$

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