Find the number of terms in the G.P. whose first term is 3 sum is `(4095)/(1024)` and the common ratio is `1/4` :

Options

A. 4
B. 5
C. 6
D. none of these

Answer

To find the number of terms in the geometric progression (G.P.) with the given parameters, we can follow these steps:

### Step 1: Write down the known values.
- First term \( a = 3 \)
- Common ratio \( r = \frac{1}{4} \)
- Sum of the first \( n \) terms \( S_n = \frac{4095}{1024} \)

### Step 2: Use the formula for the sum of the first \( n \) terms of a G.P.
Since the common ratio \( r \) is less than 1, we use the formula:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
Substituting the known values into the formula gives:
\[ \frac{4095}{1024} = \frac{3(1 - (\frac{1}{4})^n)}{1 - \frac{1}{4}} \]

### Step 3: Simplify the equation.
The denominator \( 1 - \frac{1}{4} = \frac{3}{4} \). Thus, we can rewrite the equation as:
\[ \frac{4095}{1024} = \frac{3(1 - (\frac{1}{4})^n)}{\frac{3}{4}} \]
Multiplying both sides by \( \frac{3}{4} \):
\[ \frac{4095}{1024} \cdot \frac{3}{4} = 1 - \left(\frac{1}{4}\right)^n \]

### Step 4: Calculate the left side.
Calculating \( \frac{4095 \cdot 3}{1024 \cdot 4} \):
\[ \frac{4095 \cdot 3}{4096} = 1 - \left(\frac{1}{4}\right)^n \]

### Step 5: Rearranging the equation.
This gives us:
\[ 1 - \frac{4095}{4096} = \left(\frac{1}{4}\right)^n \]
\[ \frac{1}{4096} = \left(\frac{1}{4}\right)^n \]

### Step 6: Express \( 4096 \) as a power of \( 4 \).
We know that:
\[ 4096 = 4^6 \]
Thus, we can rewrite the equation as:
\[ \frac{1}{4^6} = \left(\frac{1}{4}\right)^n \]

### Step 7: Equate the powers.
Since the bases are the same, we can equate the exponents:
\[ n = 6 \]

### Conclusion
The number of terms \( n \) in the G.P. is \( 6 \).

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