Find the geometric progression whose `5^(th)` term is 48 and `8^(th)` term is 384.

Answer

To find the geometric progression whose 5th term is 48 and 8th term is 384, we can follow these steps:

### Step 1: Define the terms of the geometric progression
In a geometric progression (GP), the nth term can be expressed as:
\[ A_n = A \cdot r^{n-1} \]
where \( A \) is the first term and \( r \) is the common ratio.

### Step 2: Write the equations for the given terms
For the 5th term:
\[ A_5 = A \cdot r^{4} = 48 \]
(This is our Equation 1)

For the 8th term:
\[ A_8 = A \cdot r^{7} = 384 \]
(This is our Equation 2)

### Step 3: Divide Equation 2 by Equation 1
To eliminate \( A \), we can divide Equation 2 by Equation 1:
\[ \frac{A \cdot r^{7}}{A \cdot r^{4}} = \frac{384}{48} \]
This simplifies to:
\[ r^{3} = \frac{384}{48} \]

### Step 4: Simplify the right-hand side
Calculating \( \frac{384}{48} \):
\[ \frac{384}{48} = 8 \]
So we have:
\[ r^{3} = 8 \]

### Step 5: Solve for \( r \)
Taking the cube root of both sides:
\[ r = \sqrt[3]{8} = 2 \]

### Step 6: Substitute \( r \) back to find \( A \)
Now that we have \( r \), we can substitute it back into Equation 1 to find \( A \):
\[ A \cdot r^{4} = 48 \]
Substituting \( r = 2 \):
\[ A \cdot (2^{4}) = 48 \]
Calculating \( 2^{4} = 16 \):
\[ A \cdot 16 = 48 \]
Now, solving for \( A \):
\[ A = \frac{48}{16} = 3 \]

### Step 7: Write the geometric progression
Now that we have both \( A \) and \( r \), we can write the GP:
- First term: \( A = 3 \)
- Second term: \( A \cdot r = 3 \cdot 2 = 6 \)
- Third term: \( A \cdot r^{2} = 3 \cdot 2^{2} = 12 \)
- Fourth term: \( A \cdot r^{3} = 3 \cdot 2^{3} = 24 \)
- Fifth term: \( A \cdot r^{4} = 3 \cdot 2^{4} = 48 \)
- Sixth term: \( A \cdot r^{5} = 3 \cdot 2^{5} = 96 \)
- Seventh term: \( A \cdot r^{6} = 3 \cdot 2^{6} = 192 \)
- Eighth term: \( A \cdot r^{7} = 3 \cdot 2^{7} = 384 \)

Thus, the geometric progression is:
\[ 3, 6, 12, 24, 48, 96, 192, 384, \ldots \]