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:

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

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

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

$a_{12} = \frac{2}{2048}$

$a_{12} = \frac{1}{1024}$