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 \( T_7 = 64 \).

### Step 1: Identify the formula for the nth term of a G.P.
The nth term of a G.P. 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 the equation for the 7th term
For the 7th term:
\[ T_7 = a \cdot r^{6} \]
Given \( T_7 = 64 \) and \( a = 729 \), we can write:
\[ 64 = 729 \cdot r^{6} \]

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

### Step 4: Simplify \( \frac{64}{729} \)
We can express \( 64 \) and \( 729 \) as powers:
\[ 64 = 2^6 \quad \text{and} \quad 729 = 3^6 \]
Thus:
\[ r^{6} = \left(\frac{2}{3}\right)^{6} \]

### Step 5: Find \( r \)
Taking the sixth root of both sides gives:
\[ r = \frac{2}{3} \]

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

### Step 8: Substitute values into the sum formula
Now substituting \( a = 729 \), \( r = \frac{2}{3} \), and \( r^7 = \frac{128}{2187} \):
\[ S_7 = \frac{729\left(1 - \frac{128}{2187}\right)}{1 - \frac{2}{3}} \]

### Step 9: Simplify the denominator
The denominator simplifies to:
\[ 1 - \frac{2}{3} = \frac{1}{3} \]

### Step 10: Simplify the expression for \( S_7 \)
Now substituting in:
\[ S_7 = \frac{729\left(1 - \frac{128}{2187}\right)}{\frac{1}{3}} = 729 \cdot 3 \left(1 - \frac{128}{2187}\right) \]
Calculating \( 1 - \frac{128}{2187} \):
\[ 1 - \frac{128}{2187} = \frac{2187 - 128}{2187} = \frac{2059}{2187} \]
Thus:
\[ S_7 = 2187 \cdot \frac{2059}{2187} = 2059 \]

### Final Answer
The sum of the first 7 terms \( S_7 \) is:
\[ \boxed{2059} \]