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:

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) 12 c^{2} - 28 c + 8 = 0 12 = 2^{2} \cdot 3 28 = 2^{2} \cdot 7 8 = 2^{3} GCD (12, 28, 8) = 2^{2} = 4 3 c^{2} - 7 c + 2 = 0 p = 3; q = - 7; r = 2 D = q^{2} - 4 p r = 7^{2} - 4 \cdot 3 \cdot 2 = 25 D > 0 c_{1, 2} = \frac{- q \pm D}{2 p} = \frac{7 \pm 2 5}{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

Our quadratic equation calculator calculates it.

q = (c_{2} + 1) / 2 = (0.3333 + 1) / 2 = \frac{2}{3} ≐ 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

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