Question

Let \( a, b, c \) be in AP and \( |a|<1, |b|<1, |c|<1 \). If \[ x = 1 + a + a^2 + a^3 + \dots \quad \text{to} \quad \infty, \] \[ y = 1 + b + b^2 + b^3 + \dots \quad \text{to} \quad \infty, \] \[ z = 1 + c + c^2 + c^3 + \dots \quad \text{to} \quad \infty, \] then \( x, y, z \) are in:

• A. AP
• B. GP
• C. HP
• D. None of these

Hint:

When terms are in arithmetic progression (AP), the corresponding sums of geometric series will form a harmonic progression (HP).

Answer 1

The Correct Option is C

Step 1: Understanding the series.
The given series for \( x, y, z \) are all geometric series of the form: \[ S = 1 + a + a^2 + a^3 + \dots = \frac{1}{1 - a}, \] \[ S = 1 + b + b^2 + b^3 + \dots = \frac{1}{1 - b}, \] \[ S = 1 + c + c^2 + c^3 + \dots = \frac{1}{1 - c}. \] These are infinite geometric series with the common ratio \( a, b, c \), respectively, and we know that the sum of a geometric series is given by \( S = \frac{1}{1 - r} \), where \( |r|<1 \).
Step 2: Determining the relationship between \( x, y, z \).
Since \( a, b, c \) are in arithmetic progression (AP), we know that: \[ b = \frac{a + c}{2}. \] This means that \( x, y, z \) are related in such a way that their reciprocals form a harmonic progression (HP). Specifically, the sums \( x, y, z \) satisfy the property of a harmonic progression, which is when the reciprocals of the terms form an arithmetic progression.
Step 3: Conclusion.
Thus, \( x, y, z \) are in harmonic progression (HP), and the correct answer is (c).