Three Numbers GP

The sum of three Numbers in geometric progression (GP) is 38, and their product is 1728. Find the Numbers.

Final Answer:

a = 8
b = 12
c = 18

s = 38 p = 1728 p = 1 2^{3} s = a + b + c p = a b c b = q \cdot a c = q b = q^{2} \cdot a s = a + q \cdot a + q^{2} \cdot a = a (1 + q + q^{2}) p = a \cdot q \cdot a \cdot q^{2} \cdot a = a^{3} \cdot q^{3} q = p^{1 / 3} / a = 12 / a s = a (1 + q + q^{2}) = 12 / q (1 + q + q^{2}) s \cdot q = 12 (1 + q + q^{2}) 38 \cdot q = 12 (1 + q + q^{2}) - 12 q^{2} + 26 q - 12 = 0 12 q^{2} - 26 q + 12 = 0 12 = 2^{2} \cdot 3 26 = 2 \cdot 13 GCD (12, 26, 12) = 2 6 q^{2} - 13 q + 6 = 0 a = 6; b = - 13; c = 6 D = b^{2} - 4 a c = 1 3^{2} - 4 \cdot 6 \cdot 6 = 25 D > 0 q_{1, 2} = \frac{- b \pm D}{2 a} = \frac{1 3 \pm 2 5}{1 2} q_{1, 2} = \frac{1 3 \pm 5}{1 2} q_{1, 2} = 1.083333 \pm 0.416667 q_{1} = 1.5 q_{2} = 0.666666667 q = q_{1} = 1.5 = \frac{3}{2} = 1 \frac{1}{2} a = 12 / q = 12 / 1.5 = 8

b = a \cdot q = 8 \cdot 1.5 = 12

c = b \cdot q = 12 \cdot 1.5 = 18 Verifying Solution: s_{2} = a + b + c = 8 + 12 + 18 = 38 p_{2} = a \cdot b \cdot c = 8 \cdot 12 \cdot 18 = 1728

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