Question

If \( a, b, c \) are in A.P. and their squares in same order form a G.P., then \( (a+c)^4 = \)

• A. \(16a^2c^2\)
• B. \(4a^2c^2\)
• C. \(8a^2c^2\)
• D. \(2a^2c^2\)
• E. \(a^2c^2\)

Hint:

Combine A.P. and G.P. conditions carefully — substitute before simplifying.

Answer 1

The Correct Option is A

Concept: If numbers are in A.P.: \[ 2b = a+c \] If squares are in G.P.: \[ b^4 = a^2 c^2 \]

Step 1: Use A.P. condition
\[ b = \frac{a+c}{2} \]

Step 2: Square both sides
\[ b^2 = \frac{(a+c)^2}{4} \]

Step 3: Use G.P. condition
\[ b^4 = a^2 c^2 \] Substitute: \[ \left(\frac{(a+c)^2}{4}\right)^2 = a^2 c^2 \] \[ \frac{(a+c)^4}{16} = a^2 c^2 \]

Step 4: Rearrangement
\[ (a+c)^4 = 16a^2 c^2 \]

Step 5: Final answer
\[ \boxed{16a^2c^2} \]