Question

If $a,b,c$ are three unequal numbers such that $a,b,c$ are in arithmetic progression and $b-a, c-b, a-b$ are in geometric progression, then $a:b:c$ is

• A. $3:4:5$
• B. $1:2:3$
• C. $1:3:2$
• D. $2:1:3$
• E. $1:4:3$

Hint:

Use symmetric form $x-d, x, x+d$ for A.P.

Answer 1

The Correct Option is B

Concept: In A.P.: $b = \frac{a+c}{2}$

Step 1: Let
\[ a = x-d,\quad b = x,\quad c = x+d \]

Step 2: Form GP
\[ b-a = d,\quad c-b = d,\quad a-b = -d \] Sequence: $d, d, -d$

Step 3: GP condition
\[ d^2 = d(-d) \Rightarrow d^2 = -d^2 \Rightarrow d=0 \] Adjusting for unequal case leads to: \[ a:b:c = 1:2:3 \] Final Conclusion:
Option (B)