Question

If $ a, a r, a r^2 $ are in a geometric progression (G.P.), then find the value of: $ \left| a \quad a r \right| \quad \left| a r^2 \quad a r^3 \right| = \left| a r^3 \quad a r^6 \right| $

• A. \( r^2 \)
• B. \( r^4 \)
• C. \( r^6 \)
• D. \( r^3 \)

Hint:

When working with geometric progressions, use the property that the ratio of consecutive terms is constant. This can simplify many problems involving sequences.

Answer 1

The Correct Option is B

Given that \( a, a r, a r^2 \) are in a geometric progression (G.P.), we know the property of a G.P. is that the ratio between consecutive terms is constant. Therefore: \[ \frac{a r}{a} = \frac{a r^2}{a r} \] Simplifying both sides: \[ r = r \] This confirms that the sequence is indeed in G.P. Now, we need to evaluate the given expression: \[ \left| a \quad a r \right| \quad \left| a r^2 \quad a r^3 \right| = \left| a r^3 \quad a r^6 \right| \] By multiplying the determinants: \[ \left| a \quad a r \right| = a^2 r, \quad \left| a r^2 \quad a r^3 \right| = a^2 r^5, \quad \left| a r^3 \quad a r^6 \right| = a^2 r^9 \]
Thus, the equation becomes: \[ a^2 r \cdot a^2 r^5 = a^2 r^9 \] This simplifies to: \[ a^4 r^6 = a^2 r^9 \] Finally, dividing both sides by \( a^2 r^6 \): \[ a^2 r^4 = a^2 r^4 \]
Thus, the correct value is \( r^4 \), which corresponds to option (B).