Mathematics

If the 4th, 10th and 16th terms of a G.P. are x, y and z respectively, prove that x, y, z are in G.P.

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Given, a₄ = x, a₁₀ = y and a₁₆ = z.

We know that
a_n = ar^{n - 1}
∴ a₄ = ar³ = x, a₁₀ = ar⁹ = y and a₁₆ = ar¹⁵ = z.

Dividing y by x we get,

$\dfrac{y}{x} = \dfrac{ar^9}{ar^3} = r^6$

Dividing z by y we get,

$\dfrac{z}{y} = \dfrac{ar^{15}}{ar^9} = r^6 \\[1em] \therefore \dfrac{y}{z} = \dfrac{z}{y}.$

Hence, proved that x, y, z are in G.P.

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