Question

In a geometric progression the difference between the fifth term and the fourth term is 576 and the difference between the second term and the first term is 9. The fifth term is:

• A. \(824\)
• B. \(768\)
• C. \(732\)
• D. \(648\)

Hint:

In GP difference problems, factor out common terms and divide equations to eliminate \(a\) and \((r-1)\).

Answer 1

The Correct Option is B

Concept: In GP: \[ a, ar, ar^2, ar^3, ar^4 \]

Step 1: Use given conditions. \[ T_5 - T_4 = ar^4 - ar^3 = ar^3(r-1)=576 \] \[ T_2 - T_1 = ar - a = a(r-1)=9 \]

Step 2: Divide equations. \[ \frac{ar^3(r-1)}{a(r-1)} = \frac{576}{9} \] \[ r^3=64 \] \[ r=4 \]

Step 3: Find \(a\). \[ a(r-1)=9 \] \[ a(3)=9 \] \[ a=3 \]

Step 4: Find fifth term. \[ T_5=ar^4 \] \[ =3\times 4^4 \] \[ =3\times 256 \] \[ =768 \] \[ \boxed{768} \]