Find the geometric progression with fourth term = 54 seventh term = 1458.

Answer

To find the geometric progression (GP) with the given fourth term and seventh term, we can follow these steps:

### Step 1: Define the General Term
The general term of a geometric progression can be expressed as:
\[ T_n = ar^{n-1} \]
where \( a \) is the first term and \( r \) is the common ratio.

### Step 2: Write the Equations for the Given Terms
From the problem, we know:
- The fourth term \( T_4 = 54 \)
- The seventh term \( T_7 = 1458 \)

Using the general term formula:
\[ T_4 = ar^{4-1} = ar^3 = 54 \quad \text{(1)} \]
\[ T_7 = ar^{7-1} = ar^6 = 1458 \quad \text{(2)} \]

### Step 3: Divide the Two Equations
To eliminate \( a \), we can divide equation (2) by equation (1):
\[ \frac{ar^6}{ar^3} = \frac{1458}{54} \]
This simplifies to:
\[ r^{6-3} = \frac{1458}{54} \]
\[ r^3 = 27 \]

### Step 4: Solve for \( r \)
Taking the cube root of both sides:
\[ r = 3 \]

### Step 5: Substitute \( r \) Back to Find \( a \)
Now that we have \( r \), we can substitute it back into equation (1) to find \( a \):
\[ ar^3 = 54 \]
Substituting \( r = 3 \):
\[ a(3^3) = 54 \]
\[ a(27) = 54 \]
\[ a = \frac{54}{27} = 2 \]

### Step 6: Write the Geometric Progression
Now that we have both \( a \) and \( r \):
- \( a = 2 \)
- \( r = 3 \)

The terms of the geometric progression can be written as:
\[ T_1 = a = 2 \]
\[ T_2 = ar = 2 \cdot 3 = 6 \]
\[ T_3 = ar^2 = 2 \cdot 3^2 = 2 \cdot 9 = 18 \]
\[ T_4 = ar^3 = 2 \cdot 27 = 54 \]
\[ T_5 = ar^4 = 2 \cdot 81 = 162 \]
\[ T_6 = ar^5 = 2 \cdot 243 = 486 \]
\[ T_7 = ar^6 = 2 \cdot 729 = 1458 \]

Thus, the geometric progression is:
\[ 2, 6, 18, 54, 162, 486, 1458 \]

### Final Answer:
The geometric progression is \( 2, 6, 18, 54, 162, 486, 1458 \).

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