Question

If \( a, b, c \) are in A.P. where \( a + b + c = 1 \) and \( a, 2b, c \) are in G.P., then the value of \( 9(a^2 + b^2 + c^2) \) is equal to:

• A. 3
• B. -3
• C. 4
• D. -4

Hint:

For problems involving A.P. and G.P., use the known properties of sequences and solve the system of equations step by step.

Answer 1

The Correct Option is B

Step 1: Use the condition for A.P.
For \( a, b, c \) to be in A.P., we know: \[ b = \frac{a+c}{2}. \] We are also given that \( a + b + c = 1 \), so substitute \( b = \frac{a+c}{2} \) into this equation to find a relationship between \( a \) and \( c \). Step 2: Use the condition for G.P.
For \( a, 2b, c \) to be in G.P., we know that the square of the middle term is the product of the first and third terms: \[ (2b)^2 = a \cdot c. \] This gives another equation to relate \( a \) and \( c \). Step 3: Solve the system of equations.
Solving these equations, we find the values for \( a \), \( b \), and \( c \), and then calculate \( 9(a^2 + b^2 + c^2) \). Final Answer: \[ \boxed{-3}. \]