if x-2, x+4 and 4x+7 are three consecutive terms of a geometric progression, find the value of the value of x and the value of the corresponding common ratio

if x = -2, the sequence is -4 , 2 , -1     common ratio: - 0.5

if x = 5, the sequence is 3 , 9 , 27     common ratio: 3

Step-by-step explanation:

aₙ = a₁ * rⁿ⁻¹     r: common ratio

a₁ = x-2    

a₂: x+4 = (x-2) * r²⁻¹ = (x-2) * r   ...(1)

a₃: 4x+7 = (x-2) * r³⁻¹ = (x-2) * r²   ...(2)

(a₂)² : (x+4)² = (x-2)² * r²   ... (3)

(3)/(2) : (x+4)² / (4x + 7) = (x-2)² * r² / (x-2) * r² = (x-2) / 1

(x+4)² = (4x + 7)*(x-2)

x² + 8x + 16 = 4x² - x - 14

3x² - 9x -30 = 0     ... (4)  

(4)/3 : x² - 3x - 10 = (x+2) (x-5) = 0

x = -2      or     x = 5

if x = -2, the sequence is -4 , 2 , -1     common ratio: - 0.5

if x = 5, the sequence is 3 , 9 , 27     common ratio: 3

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step-by-step explanation: