GP - three members

The second and third of a geometric progression are 24 and 12(c+1), respectively, given that the sum of the first three terms of progression is 76. determine the value of c.

Correct answer:

c₁ =  2
c₂ =  0.3333

Step-by-step explanation:

$$\begin{gathered} a_2=24 \\ {a}_3=12(c+1) \\ q=a_3/a_2=12(c+1)/24=(c+1)/2 \\ {a}_1=a_2/q=24\cdot 2/(c+1)=48/(c+1) \\ s=a_1+a_2+a_3=76 \\ 48/(c+1)+24+12(c+1)=76 \\ 48+24(c+1)+12(c+1{)}^2=76(c+1) \\ 12c^2-28c+8=0 \\ 12=2^2\cdot 3 \\ 28=2^2\cdot 7 \\ 8=2^3 \\ \text{GCD}(12,28,8)=2^2=4 \\ 3c^2-7c+2=0 \\ p=3;q=-7;r=2 \\ D=q^2-4pr=7^2-4\cdot 3\cdot 2=25 \\ D>0 \\ {c}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{7\pm \sqrt{25}}{6} \\ {c}_{1,2}=\frac{7\pm 5}{6} \\ {c}_{1,2}=1.166667\pm 0.833333 \\ {c}_1=2 \\ {c}_2=0.333333333 \\ c=c_1=2 \\ q=(c+1)/2=(2+1)/2=\frac{3}{2}=1\frac{1}{2}=1.5 \\ {a}_1=a_2/q=24/1.5=16 \\ {a}_3=a_2\cdot q=24\cdot 1.5=36 \\ {s}_2=a_1+a_2+a_3=16+24+36=76 \\ {s}_2=s \\ {c}_1=2 \end{gathered}$$

Our quadratic equation calculator calculates it.

$$\begin{gathered} q=(c_2+1)/2=(0.3333+1)/2=\frac{2}{3}\doteq 0.6667 \\ {a}_{11}=a_2/q=24/0.6667=36 \\ {a}_{33}=a_2\cdot q=24\cdot 0.6667=16 \\ {s}_3=a_{11}+a_2+a_{33}=36+24+16=76 \\ {s}_3=s_2=s \\ {c}_2=0.3333 \end{gathered}$$

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