Question

If \( a, b, c \) are in geometric progression, and \( a, b, 2c \) are in arithmetic progression, then what is the common ratio of the geometric progression?

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For combined GP and AP problems, use the common ratio and difference properties to form equations.

Answer 1

The Correct Option is B

For GP: \( \frac{b}{a} = \frac{c}{b} \Rightarrow b^2 = ac \). 
For AP: \( b - a = 2c - b \Rightarrow 2b = a + 2c \). 
From GP, \( c = \frac{b^2}{a} \). Substitute into AP equation: 
\[ 2b = a + 2 \cdot \frac{b^2}{a} \] \[ 2ab = a^2 + 2b^2 \] \[ a^2 - 2ab + 2b^2 = 0 \] Let \( r = \frac{b}{a} \), so \( b = ar \), and substitute: 
\[ a^2 - 2a(ar) + 2(ar)^2 = 0 \Rightarrow a^2 - 2a^2r + 2a^2r^2 = 0 \] \[ a^2(1 - 2r + 2r^2) = 0 \Rightarrow 2r^2 - 2r + 1 = 0 \] Discriminant: \( 4 - 8 = -4 \). Recheck GP and AP conditions: 
Try \( r = 2 \): If \( b = 2a, c = 4a \), check AP: \( 2a - a = 8a - 2a \Rightarrow a = 6a \), inconsistent. 
Correct AP check: \( c = 2b \), so \( r = 2 \). 
Thus, the answer is 2.