Question

The first and the twentieth terms of a G.P. are 512 and \(\frac{1}{1024}\) respectively. Then the common ratio is

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

Hint:

When dealing with terms that are powers of 2 (like 512 and 1024), always rewrite them in the form \(2^k\) to simplify calculations involving exponents.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We are given specific terms of a Geometric Progression (G.P.). We use the formula for the \(n^{th}\) term of a G.P. to find the common ratio \(r\).

Step 2: Key Formula or Approach:
The \(n^{th}\) term of a G.P. is \(a_n = a r^{n-1}\), where \(a\) is the first term and \(r\) is the common ratio.

Step 3: Detailed Explanation:
Given:
First term \(a = 512 = 2^9\).
Twentieth term \(a_{20} = \frac{1}{1024} = 2^{-10}\).
Using the formula for the 20th term:
\[ a_{20} = a r^{20-1} = a r^{19} \]
\[ 2^{-10} = 2^9 \cdot r^{19} \]
Divide both sides by \(2^9\):
\[ r^{19} = \frac{2^{-10}}{2^9} = 2^{-19} \]
\[ r^{19} = (2^{-1})^{19} \]
Taking the 19th root on both sides:
\[ r = 2^{-1} = \frac{1}{2} \]

Step 4: Final Answer:
The common ratio is \(\frac{1}{2}\).