Question

If the 2^nd term and 5^th term of a GP are 24, 81 then the r=

• A. 16
• B. 3
• C. 20
• D. \(\frac{3}{2}\)

Answer 1

The Correct Option is D

Let the first term of the G.P. be \( a \) and the common ratio be \( r \). The 2nd term is \( a r \) and the 5th term is \( a r^4 \). Given: \[ a r = 24 \quad \text{(1)} \] \[ a r^4 = 81 \quad \text{(2)} \] Divide equation (2) by equation (1): \[ \frac{a r^4}{a r} = \frac{81}{24} \Rightarrow r^3 = \frac{81}{24} = \frac{27}{8} \Rightarrow r = \sqrt[3]{\frac{27}{8}} = \frac{3}{2} \] So, the correct value of \( r \) is \(\frac{3}{2}\). Wait! But this gives us option (4), not option (2). Let's double-check by solving precisely. Let’s verify: Given: 2nd term = \( ar = 24 \) 5th term = \( ar^4 = 81 \) Divide: \[ \frac{ar^4}{ar} = r^3 = \frac{81}{24} = \frac{27}{8} \Rightarrow r = \left( \frac{27}{8} \right)^{1/3} = \frac{3}{2} \] Confirmed again, \( r = \frac{3}{2} \)

The correct option is (D): \(\frac{3}{2}\)