If x = _n=0^∞ a^n, y = _n=0^∞ b^n, z = _n=0^∞ c^n where a, b, c are in A.P. and |a| < 1, |b|<1, |c|< 1 , then x, y, z are in: - Mathematics MCQ AIEEE 2026 | ExamTest.live

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If x = \sum_{n = 0}^{\infty} a^{n}, y = \sum_{n = 0}^{\infty} b^{n}, z = \sum_{n = 0}^{\infty} c^{n} where a, b, c are in A.P. and ∣ a ∣ < 1, ∣ b ∣ < 1, ∣ c ∣ < 1, then x, y, z are in:

Solution & Step-by-step Explanation

The infinite geometric series are: x = \frac{1}{1 - a} ⟹ 1 - a = \frac{1}{x} ⟹ a = 1 - \frac{1}{x} y = \frac{1}{1 - b} ⟹ 1 - b = \frac{1}{y} ⟹ b = 1 - \frac{1}{y} z = \frac{1}{1 - c} ⟹ 1 - c = \frac{1}{z} ⟹ c = 1 - \frac{1}{z} Since a, b, c are in A.P.: 2 b = a + c 2 (1 - \frac{1}{y}) = (1 - \frac{1}{x}) + (1 - \frac{1}{z}) 2 - \frac{2}{y} = 2 - \frac{1}{x} - \frac{1}{z} \frac{2}{y} = \frac{1}{x} + \frac{1}{z} This is the condition for x, y, z to be in Harmonic Progression (H.P.).

Try it yourself before checking the explanation above.

If x = \sum_{n = 0}^{\infty} a^{n}, y = \sum_{n = 0}^{\infty} b^{n}, z = \sum_{n = 0}^{\infty} c^{n} where a, b, c are in A.P. and ∣ a ∣ < 1, ∣ b ∣ < 1, ∣ c ∣ < 1, then x, y, z are in:

G.P.

A.P.

Arithmetic-Geometric Progression

H.P.

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