Geometric progression

Determine the quotient and the first member of GP if a3 = 0.39 and a1 + a2 = 0.39.

Final Answer:

q₁ = 1.618
q₂ = -0.618
a₁ = 0.149
b₁ = 1.021

a_{1} + a_{1} q = 0.39 a_{1} (1 + q) = 0.39 a_{1} = 0.39 / (1 + q) a_{1} \cdot q^{2} = 0.39 0.39 / (1 + q) \cdot q^{2} = 0.39 q^{2} = 1 + q q^{2} - q - 1 = 0 q^{2} - q - 1 = 0 a = 1; b = - 1; c = - 1 D = b^{2} - 4 a c = 1^{2} - 4 \cdot 1 \cdot (- 1) = 5 D > 0 q_{1, 2} = \frac{- b \pm D}{2 a} = \frac{1 \pm 5}{2} q_{1, 2} = 0.5 \pm 1.118034 q_{1} = 1.618033989 q_{2} = - 0.618033989 q_{1} = 1.618

q_{2} = (- 0.618) = - 0.618

a_{1} = 0.39 / (1 + q_{1}) = 0.39 / (1 + 1.618) ≐ 0.149 a_{2} = q_{1} \cdot a_{1} = 1.618 \cdot 0.149 ≐ 0.241 a_{3} = q_{1} \cdot a_{2} = 1.618 \cdot 0.241 = \frac{3 9}{1 0 0} = 0.39 s = a_{1} + a_{2} = 0.149 + 0.241 = \frac{3 9}{1 0 0} = 0.39 a_{1} = 0.149

b_{1} = 0.39 / (1 + q_{2}) = 0.39 / (1 + (- 0.618)) ≐ 1.021 b_{2} = q_{2} \cdot b_{1} = (- 0.618) \cdot 1.021 ≐ - 0.631 b_{3} = q_{2} \cdot b_{2} = (- 0.618) \cdot (- 0.631) = \frac{3 9}{1 0 0} = 0.39 s = b_{1} + b_{2} = 1.021 + (- 0.631) = \frac{3 9}{1 0 0} = 0.39 b_{1} = 1.021

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