Question

If \( a, b, c \) are in AP, \( b - a, c - b \) and \( a \) are in GP, then \( a : b : c \) is:

• A. \( 1 : 2 : 3 \)
• B. \( 1 : 3 : 5 \)
• C. \( 2 : 3 : 4 \)
• D. \( 1 : 2 : 4 \)

Hint:

When solving for ratios in AP and GP, express terms in terms of a common variable and use the relations between the sequences to simplify.

Answer 1

The Correct Option is A

We are given that \( a, b, c \) are in arithmetic progression (AP) and \( b - a, c - b, a \) are in geometric progression (GP). Step 1: Express the terms in terms of \( a \) and \( d \) Since \( a, b, c \) are in AP, we know that: \[ b = a + d, \quad c = a + 2d \] Next, since \( b - a, c - b, a \) are in GP, we know that: \[ \frac{c - b}{b - a} = \frac{a}{c - b} \] Substitute \( b = a + d \) and \( c = a + 2d \): \[ \frac{(a + 2d) - (a + d)}{(a + d) - a} = \frac{a}{(a + 2d) - (a + d)} \] Simplifying the equation: \[ \frac{d}{d} = \frac{a}{d} \] This simplifies to \( a = d \). Step 2: Find the ratio \( a : b : c \) Substitute \( d = a \) into the expressions for \( b \) and \( c \): \[ b = a + a = 2a, \quad c = a + 2a = 3a \] Thus, the ratio is: \[ a : b : c = 1 : 2 : 3 \] Thus, the correct answer is \( 1 : 2 : 3 \).