If x, 2y, 3z are in AP where the distinct numbers x, y, z are in GP, find the common - Maths - Sequences and Series

Question

If x, 2y, 3z are in AP where the distinct numbers x, y, z are in
GP, find the common ratio of the GP

Vijay Kumar Gupta · Accepted Answer

Dear student
The solution to your 1st query has been provided below:
Please find below the solution to the asked query:

Since x, 2 y, 3 z are in AP, this implies that, 2 y - x = 3 z - 2 y 2 y + 2 y = x + 3 z 4 y = x + 3 z . . . . . (1) Also, it is given that x, y, z are in GP which implies that, \frac{y}{x} = \frac{z}{y} = r, Here r is common ratio of GP y^{2} = xz . . . . . (2) Take square on both sides of (1) {(4 y)}^{2} = {(x + 3 z)}^{2} 16 y^{2} = x^{2} + 9 z^{2} + 6 x z 16 xz = x^{2} + 9 z^{2} + 6 x z (Since y^{2} = xz from (2)) 0 = x^{2} + 9 z^{2} + 6 z - 16 xz 0 = x^{2} - 10 xz + 9 z^{2} This gives, x^{2} - 10 xz + 9 z^{2} = 0 x^{2} - 9 xz - xz + 9 z^{2} = 0 x (x - 9 z) - z (x - 9 z) = 0 (x - 9 z) (x - z) = 0 Note that x \neq z \Rightarrow x - z \neq 0 This gives, x - 9 z = 0 x = 9 z From (1), we have 4 y = 9 z + 3 z 4 y = 12 z y = 3 z Thus the common ratio is given by, r = \frac{y}{x} = \frac{3 z}{9 z} = \frac{1}{3}

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If x, 2y, 3z are in AP where the distinct numbers x, y, z are in GP, find the common ratio of the GP.