Question

The second term of a G.P. is \( \frac{1}{2} \). If the product of first five terms is 32, then the common ratio of the G.P. is:

• A. \( \frac{1}{4} \)
• B. \( 4 \)
• C. \( \frac{1}{8} \)
• D. \( 8 \)
• E. \( \frac{1}{2} \)

Hint:

For problems involving geometric progressions, the product of the first \( n \) terms can be expressed as \( a^n r^{\frac{n(n-1)}{2}} \), and the common ratio can often be found by solving equations involving the terms.

Answer 1

The Correct Option is B

Let the first term of the geometric progression (G.P.) be \( a \), and the common ratio be \( r \). 
From the given information: - The second term is \( \frac{1}{2} \). In terms of \( a \) and \( r \), this can be written as: \[ a r = \frac{1}{2} \] Hence, the first term is: \[ a = \frac{1}{2r} \] - The product of the first five terms is 32. The product of the first five terms of a G.P. is given by: \[ a \cdot a r \cdot a r^2 \cdot a r^3 \cdot a r^4 = a^5 r^{10} \] Using the given value: \[ a^5 r^{10} = 32 \] Substituting \( a = \frac{1}{2r} \) into this equation: \[ \left(\frac{1}{2r}\right)^5 r^{10} = 32 \] Simplifying: \[ \frac{1}{(2r)^5} \cdot r^{10} = 32 \] \[ \frac{r^5}{32 r^5} = 32 \] \[ \frac{1}{32} \cdot r^5 = 32 \] \[ r^5 = 1024 \] Taking the fifth root of both sides: \[ r = 4 \] Thus, the common ratio of the G.P. is \( 4 \).
Thus, the correct answer is option (B), \( 4 \).