Geometric sequence sum

Determine s5 of the geometric sequence if: a1 + a2 = 10 and a4 - a2 = 120

Final Answer:

s₅ = 682

a_{1} + a_{2} = 10 a_{4} - a_{2} = 120 a_{2} = a_{1} \cdot q a_{4} = q^{2} \cdot a_{2} = q^{3} a_{1} a_{1} + q \cdot a_{1} = 10 q^{3} a_{1} - q \cdot a_{1} = 120 a_{1} (1 + q) = 10 a_{1} (q^{3} - q) = 120 a_{1} (1 + q) = 10 a_{1} q (q^{2} - 1) = 120 a_{1} (1 + q) = 10 a_{1} q (q - 1) (q + 1) = 120 a_{1} (1 + q) = 10 a_{1} (q + 1) \cdot (q (q - 1)) = 120 10 \cdot (q (q - 1)) = 120 10 \cdot (q (q - 1)) = 120 10 q^{2} - 10 q - 120 = 0 10 = 2 \cdot 5 120 = 2^{3} \cdot 3 \cdot 5 GCD (10, 10, 120) = 2 \cdot 5 = 10 q^{2} - q - 12 = 0 a = 1; b = - 1; c = - 12 D = b^{2} - 4 a c = 1^{2} - 4 \cdot 1 \cdot (- 12) = 49 D > 0 q_{1, 2} = \frac{- b \pm D}{2 a} = \frac{1 \pm 4 9}{2} q_{1, 2} = \frac{1 \pm 7}{2} q_{1, 2} = 0.5 \pm 3.5 q_{1} = 4 q_{2} = - 3 q = q_{1} = 4 a_{1} = 10 / (1 + q) = 10 / (1 + 4) = 2 a_{2} = q \cdot a_{1} = 4 \cdot 2 = 8 a_{3} = q \cdot a_{2} = 4 \cdot 8 = 32 a_{4} = q \cdot a_{3} = 4 \cdot 32 = 128 a_{5} = q \cdot a_{4} = 4 \cdot 128 = 512 s_{5} = a_{1} + a_{2} + a_{3} + a_{4} + a_{5} = 2 + 8 + 32 + 128 + 512 = 682

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