Question

If \( 1, a, b, c, 16 \) are in geometric progression, then \( \sqrt[3]{abc} \) is equal to

• A. \( 1 \)
• B. \( 2 \)
• C. \( 6 \)
• D. \( 4 \)
• E. \( 8 \)

Hint:

In GP, expressing terms as powers of the common ratio \( r \) makes multiplication and root calculations very simple.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem involves a finite geometric progression (GP). We can solve it either by finding the common ratio and then the individual terms, or by using the properties of a GP.
Step 2: Key Formula or Approach:
Method 1: Finding the Common Ratio (r)
Let the GP be \(a_1, a_2, a_3, a_4, a_5\). We have \(a_1=1\) and \(a_5=16\).
The formula for the n-th term is \(a_n = a_1 \cdot r^{n-1}\).
Method 2: Using Properties of a GP
In a finite GP, the product of terms equidistant from the beginning and the end is constant and equals the product of the first and last terms. Also, if the number of terms is odd, the middle term squared is equal to this product.
Step 3: Detailed Explanation:
Applying Method 1:
We have \(a_5 = a_1 \cdot r^{5-1}\).
\[ 16 = 1 \cdot r^4 \] \[ r^4 = 16 \] Assuming the terms are positive, we take the positive real root: \(r = 2\).
Now we can find the terms a, b, and c:
\(a = a_2 = a_1 \cdot r = 1 \cdot 2 = 2\)
\(b = a_3 = a_2 \cdot r = 2 \cdot 2 = 4\)
\(c = a_4 = a_3 \cdot r = 4 \cdot 2 = 8\)
Now, calculate the required value:
\[ \sqrt[3]{abc} = \sqrt[3]{2 \cdot 4 \cdot 8} = \sqrt[3]{64} = 4 \] Applying Method 2:
The GP is 1, a, b, c, 16.
The product of the first and last term is \(1 \times 16 = 16\).
Due to the symmetry property:
The product of the second and fourth terms is equal to the product of the first and fifth terms: \(a \cdot c = 1 \cdot 16 = 16\).
The square of the middle term (b) is also equal to this product: \(b^2 = 1 \cdot 16 = 16 \implies b = 4\).
Now, we can find the product \(abc\):
\[ abc = (ac) \cdot b = 16 \cdot 4 = 64 \] Finally, calculate the cube root:
\[ \sqrt[3]{abc} = \sqrt[3]{64} = 4 \] Step 4: Final Answer:
The value of \(\sqrt[3]{abc}\) is 4. Therefore, option (D) is the correct answer.