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

Step 1: Understanding the Concept:
This problem is another application of the 'middle term' property of a Geometric Progression (G.P.). For an odd number of terms, the product is equal to the middle term raised to the power of the number of terms.
Step 2: Key Formula or Approach:
Let the first five terms of the G.P. be `a_1, a_2, a_3, a_4, a_5`. A convenient way to represent these terms is centered around the middle term `a_3` (let's call it `a` for simplicity): Terms: \(\frac{a}{r^2}, \frac{a}{r}, a, ar, ar^2\). The product of these terms is \(\left(\frac{a}{r^2}\right) \cdot \left(\frac{a}{r}\right) \cdot a \cdot (ar) \cdot (ar^2) = a^5\). We are given that this product is 32.
Step 3: Detailed Explanation:
Let `a_3` be the third term of the G.P. The product of the first 5 terms is `a_1 \cdot a_2 \cdot a_3 \cdot a_4 \cdot a_5`. Using the property mentioned above, this product is equal to `(a_3)^5`.
We are given that the product is 32.
\[ (a_3)^5 = 32 \] To solve for `a_3`, we need to find the fifth root of 32. We know that `2^5 = 32`.
\[ (a_3)^5 = 2^5 \] Therefore, the third term `a_3` is 2.
The condition that the terms are "increasing" means the common ratio `r>1`. This information ensures that the G.P. is well-defined but is not needed to find the value of the third term.
Step 4: Final Answer:
The third term of the G.P. is 2.