Question

The 12^th term of the geometric progression (G.P.) \(2,1,\frac 12, \frac 14, \frac 18,…….\) is

• A. \(\frac {1}{2^9}\)
• B. \(\frac {1}{2^8}\)
• C. \(\frac {1}{2^{11}}\)
• D. \(\frac {1}{2^{10}}\)

Answer 1

The Correct Option is D

1. The given geometric progression (G.P.) is:

2, 1, \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{8}\), . . .

2. The formula for the \(n\)th term of a G.P. is:

\( a_n = a_1 \cdot r^{(n-1)} \)

Where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

3. Finding the common ratio:

The common ratio \( r \) is the ratio of any term to the previous term:

\( r = \frac{1}{2} \div 1 = \frac{1}{2} \)

4. Finding the 12th term:

Substitute \( a_1 = 2 \), \( r = \frac{1}{2} \), and \( n = 12 \) into the formula:

\( a_{12} = 2 \cdot \left(\frac{1}{2}\right)^{(12-1)} \)

\( a_{12} = 2 \cdot \left(\frac{1}{2}\right)^{11} \)

\( a_{12} = 2 \cdot \frac{1}{2^{11}} \)

\( a_{12} = \frac{2}{2048} \)

\( a_{12} = \frac{1}{1024} \)