Given a G.P with a=729 and 7th term 64,determine `S_(7)`.

Explanation

To solve the problem step-by-step, we need to find the sum of the first 7 terms of a geometric progression (G.P.) where the first term \( a = 729 \) and the 7th term \( A_7 = 64 \).

### Step 1: Understand the formula for the nth term of a G.P.
The nth term of a G.P. can be expressed as:
\[ A_n = a \cdot r^{n-1} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

### Step 2: Set up the equation for the 7th term
Given that \( A_7 = 64 \), we can write:
\[ A_7 = a \cdot r^{7-1} = a \cdot r^6 \]
Substituting the known values:
\[ 64 = 729 \cdot r^6 \]

### Step 3: Solve for \( r^6 \)
Rearranging the equation gives:
\[ r^6 = \frac{64}{729} \]

### Step 4: Simplify \( r^6 \)
Recognizing that \( 64 = 2^6 \) and \( 729 = 3^6 \):
\[ r^6 = \left(\frac{2}{3}\right)^6 \]
Taking the sixth root on both sides, we find:
\[ r = \frac{2}{3} \]

### Step 5: Use the formula for the sum of the first n terms of a G.P.
The sum of the first \( n \) terms \( S_n \) of a G.P. is given by:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
For \( n = 7 \):
\[ S_7 = \frac{729(1 - r^7)}{1 - r} \]

### Step 6: Calculate \( r^7 \)
We need to find \( r^7 \):
\[ r^7 = \left(\frac{2}{3}\right)^7 = \frac{2^7}{3^7} = \frac{128}{2187} \]

### Step 7: Substitute values into the sum formula
Now substituting \( r \) and \( r^7 \) into the sum formula:
\[ S_7 = \frac{729 \left(1 - \frac{128}{2187}\right)}{1 - \frac{2}{3}} \]

### Step 8: Simplify the denominator
Calculating the denominator:
\[ 1 - \frac{2}{3} = \frac{1}{3} \]

### Step 9: Simplify the numerator
Now simplify the numerator:
\[ 1 - \frac{128}{2187} = \frac{2187 - 128}{2187} = \frac{2059}{2187} \]

### Step 10: Combine the results
Putting it all together:
\[ S_7 = \frac{729 \cdot \frac{2059}{2187}}{\frac{1}{3}} = 729 \cdot \frac{2059}{2187} \cdot 3 \]

### Step 11: Calculate the final value
Calculating \( 729 \cdot 3 = 2187 \):
\[ S_7 = \frac{2187 \cdot 2059}{2187} = 2059 \]

Thus, the sum of the first 7 terms \( S_7 \) is:
\[ \boxed{2059} \]