Which term of the G.P. 2,1,`1/2`,`/4`…………is `1/(1024)`?

Answer

To find which term of the geometric progression (G.P.) 2, 1, 1/2, 1/4, ... is equal to \( \frac{1}{1024} \), we can follow these steps:

### Step 1: Identify the first term and common ratio
The first term \( a \) of the G.P. is:
\[ a = 2 \]

To find the common ratio \( r \), we can divide the second term by the first term:
\[ r = \frac{1}{2} \div 2 = \frac{1}{2} \]

### Step 2: Write the formula for the nth term of a G.P.
The nth term \( T_n \) of a G.P. can be expressed as:
\[ T_n = a \cdot r^{n-1} \]
Substituting the values of \( a \) and \( r \):
\[ T_n = 2 \cdot \left(\frac{1}{2}\right)^{n-1} \]

### Step 3: Set the nth term equal to \( \frac{1}{1024} \)
We need to find \( n \) such that:
\[ 2 \cdot \left(\frac{1}{2}\right)^{n-1} = \frac{1}{1024} \]

### Step 4: Simplify the equation
Dividing both sides by 2:
\[ \left(\frac{1}{2}\right)^{n-1} = \frac{1}{2048} \]

### Step 5: Express \( \frac{1}{2048} \) as a power of \( \frac{1}{2} \)
We know that:
\[ 2048 = 2^{11} \quad \text{thus} \quad \frac{1}{2048} = \frac{1}{2^{11}} = \left(\frac{1}{2}\right)^{11} \]

### Step 6: Set the exponents equal to each other
Since the bases are the same, we can equate the exponents:
\[ n - 1 = 11 \]

### Step 7: Solve for \( n \)
Adding 1 to both sides gives:
\[ n = 12 \]

### Conclusion
The term of the G.P. that is equal to \( \frac{1}{1024} \) is the 12th term.