Determine the number n of terms of the GP 3,6,12,…….. So that `S_(n)=381`

Answer

To determine the number \( n \) of terms of the geometric progression (GP) \( 3, 6, 12, \ldots \) such that the sum of the first \( n \) terms \( S_n = 381 \), we can follow these steps:

### Step 1: Identify the first term and common ratio
The first term \( a \) of the GP is \( 3 \) and the common ratio \( r \) can be calculated as:
\[ r = \frac{\text{second term}}{\text{first term}} = \frac{6}{3} = 2 \]

### Step 2: Write the formula for the sum of the first \( n \) terms of a GP
The formula for the sum of the first \( n \) terms of a GP is given by:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \quad \text{(for } r \neq 1\text{)} \]
Substituting the values of \( a \) and \( r \):
\[ S_n = \frac{3(1 - 2^n)}{1 - 2} = \frac{3(1 - 2^n)}{-1} = 3(2^n - 1) \]

### Step 3: Set up the equation for the sum
We know that \( S_n = 381 \). Therefore, we set up the equation:
\[ 3(2^n - 1) = 381 \]

### Step 4: Solve for \( 2^n \)
Dividing both sides by \( 3 \):
\[ 2^n - 1 = \frac{381}{3} \]
Calculating the right side:
\[ 2^n - 1 = 127 \]
Adding \( 1 \) to both sides:
\[ 2^n = 128 \]

### Step 5: Solve for \( n \)
Now, we can express \( 128 \) as a power of \( 2 \):
\[ 2^n = 2^7 \]
Thus, by comparing the exponents, we find:
\[ n = 7 \]

### Conclusion
The number of terms \( n \) in the GP such that \( S_n = 381 \) is:
\[ \boxed{7} \]