find the G.P. whose `5^(th)` term is 48 and `8^(th)` term is 384.

Answer

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

### Step 1: Understand the terms of a G.P.
In a geometric progression, the nth term can be expressed as:
\[ T_n = A \cdot R^{n-1} \]
where \( A \) is the first term, \( R \) is the common ratio, and \( n \) is the term number.

### Step 2: Write equations for the given terms
From the problem, we know:
- The 5th term \( T_5 = A \cdot R^{4} = 48 \) (1)
- The 8th term \( T_8 = A \cdot R^{7} = 384 \) (2)

### Step 3: Set up the equations
From equation (1):
\[ A \cdot R^{4} = 48 \]
From equation (2):
\[ A \cdot R^{7} = 384 \]

### Step 4: Divide the second equation by the first
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:
\[ R^{3} = 8 \]

### Step 5: Solve for \( R \)
Taking the cube root of both sides:
\[ R = 2 \]

### Step 6: Substitute \( R \) back to find \( A \)
Now, substitute \( R = 2 \) back into equation (1):
\[ A \cdot (2^{4}) = 48 \]
This simplifies to:
\[ A \cdot 16 = 48 \]
Now, divide both sides by 16:
\[ A = \frac{48}{16} = 3 \]

### Step 7: Write the G.P.
Now that we have \( A = 3 \) and \( R = 2 \), we can write the G.P.:
The terms of the G.P. are:
- 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 G.P. is:
\[ 3, 6, 12, 24, 48, 96, 192, 384, \ldots \]

### Final Answer:
The G.P. is \( 3, 6, 12, 24, 48, 96, 192, 384, \ldots \)

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