The first term of an arithmetic progression is `1` and the sum of the first nine terms equal to `369`. The first and the ninth term of a geometric progression coincide with the first and the ninth term of the arithmetic progression. Find the seventh term of the geometric progression.

Explanation

To solve the problem step by step, we will follow the information provided about the arithmetic progression (AP) and geometric progression (GP).

### Step 1: Identify the first term and the sum of the first nine terms of the AP.
- The first term \( A \) of the AP is given as \( 1 \).
- The sum of the first nine terms \( S_9 \) is given as \( 369 \).

### Step 2: Use the formula for the sum of the first \( n \) terms of an AP.
The formula for the sum of the first \( n \) terms of an AP is:
\[ S_n = \frac{n}{2} \times (2A + (n-1)D) \]
For our case, \( n = 9 \), \( A = 1 \), and \( S_9 = 369 \):
\[ 369 = \frac{9}{2} \times (2 \times 1 + (9-1)D) \]

### Step 3: Simplify the equation.
Multiplying both sides by \( 2 \) to eliminate the fraction:
\[ 738 = 9 \times (2 + 8D) \]
Dividing both sides by \( 9 \):
\[ 82 = 2 + 8D \]

### Step 4: Solve for \( D \).
Subtract \( 2 \) from both sides:
\[ 80 = 8D \]
Dividing by \( 8 \):
\[ D = 10 \]

### Step 5: Find the ninth term of the AP.
The ninth term \( T_9 \) of the AP can be calculated using the formula:
\[ T_n = A + (n-1)D \]
For \( n = 9 \):
\[ T_9 = 1 + (9-1) \times 10 = 1 + 80 = 81 \]

### Step 6: Relate the first and ninth terms of the GP to the AP.
The first term of the GP \( G_1 \) is \( 1 \) (same as the first term of the AP).
The ninth term of the GP \( G_9 \) is given by:
\[ G_9 = G_1 \times R^8 = 1 \times R^8 \]
Since \( G_9 \) also equals the ninth term of the AP, we have:
\[ R^8 = 81 \]

### Step 7: Solve for \( R \).
Taking the eighth root of both sides:
\[ R = 81^{1/8} = 3^{4/8} = 3^{1/2} = \sqrt{3} \]

### Step 8: Find the seventh term of the GP.
The seventh term \( G_7 \) of the GP is given by:
\[ G_7 = G_1 \times R^6 = 1 \times (\sqrt{3})^6 = 3^3 = 27 \]

### Final Answer:
The seventh term of the geometric progression is \( 27 \).

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