Question

The product of first 5 terms of a G.P., whose terms are increasing, is 32. The third term of the G.P. is

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

Hint:

Product of odd number of G.P. terms can be written as power of middle term.

Answer 1

The Correct Option is A

Concept: In G.P., terms are: \[ a, ar, ar^2, ar^3, ar^4 \]

Step 1: Form product of 5 terms.
\[ a \cdot ar \cdot ar^2 \cdot ar^3 \cdot ar^4 = a^5 r^{10} \] \[ a^5 r^{10} = 32 \quad ...(1) \]

Step 2: Group terms.
\[ (a r^2)^5 = 32 \]

Step 3: Solve.
\[ (ar^2)^5 = 32 = 2^5 \Rightarrow ar^2 = 2 \]

Step 4: Interpret result.
\[ \text{Third term} = ar^2 = 2 \] Since terms are increasing, \( r>1 \), so positive value is valid.