If the first term in a geometric sequence is 3, and if the third term is 48, what is the 11th term ?

Options

A. 228
B. 528
C. `110,592`
D. `3,145,728`

Explanation

To find the 11th term of the geometric sequence where the first term is 3 and the third term is 48, we can follow these steps:

### Step 1: Identify the first term and the third term
The first term \( A \) is given as:
\[ A = 3 \]
The third term \( T_3 \) is given as:
\[ T_3 = 48 \]

### Step 2: Use the formula for the nth term of a geometric sequence
The nth term of a geometric sequence can be expressed as:
\[ T_n = A \cdot R^{n-1} \]
For the third term:
\[ T_3 = A \cdot R^{3-1} = A \cdot R^2 \]
Substituting the known values:
\[ 48 = 3 \cdot R^2 \]

### Step 3: Solve for \( R^2 \)
To isolate \( R^2 \), divide both sides by 3:
\[ R^2 = \frac{48}{3} = 16 \]

### Step 4: Find \( R \)
Now, take the square root of both sides to find \( R \):
\[ R = \sqrt{16} = 4 \]

### Step 5: Find the 11th term
Now that we have both \( A \) and \( R \), we can find the 11th term using the formula:
\[ T_{11} = A \cdot R^{11-1} = A \cdot R^{10} \]
Substituting the values:
\[ T_{11} = 3 \cdot 4^{10} \]

### Step 6: Calculate \( 4^{10} \)
We can calculate \( 4^{10} \):
\[ 4^{10} = (2^2)^{10} = 2^{20} \]
Calculating \( 2^{20} \):
\[ 2^{20} = 1048576 \]
Thus,
\[ 4^{10} = 1048576 \]

### Step 7: Final Calculation
Now substitute back to find \( T_{11} \):
\[ T_{11} = 3 \cdot 1048576 = 3145728 \]

### Conclusion
The 11th term of the geometric sequence is:
\[ T_{11} = 3145728 \]