Find the sum of 7 terms of the series `(16)/(27)-(8)/(9)+(4)/(3)-`….

Explanation

To find the sum of the first 7 terms of the series \( \frac{16}{27} - \frac{8}{9} + \frac{4}{3} - \ldots \), we can identify the series as a geometric progression (GP).

### Step 1: Identify the first term and common ratio
The first term \( a \) of the series is:
\[ a = \frac{16}{27} \]

To find the common ratio \( r \), we can take the second term and divide it by the first term:
\[ r = \frac{-\frac{8}{9}}{\frac{16}{27}} = -\frac{8}{9} \times \frac{27}{16} = -\frac{8 \times 27}{9 \times 16} = -\frac{3}{2} \]

### Step 2: Use the formula for the sum of the first n terms of a GP
The formula for the sum of the first \( n \) terms of a geometric progression is given by:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
where \( n \) is the number of terms.

In our case, we want to find \( S_7 \):
\[ S_7 = \frac{16}{27} \cdot \frac{(-\frac{3}{2})^7 - 1}{-\frac{3}{2} - 1} \]

### Step 3: Calculate \( r^7 \)
First, calculate \( (-\frac{3}{2})^7 \):
\[ (-\frac{3}{2})^7 = -\frac{3^7}{2^7} = -\frac{2187}{128} \]

### Step 4: Substitute values into the formula
Now substitute \( r^7 \) into the formula:
\[ S_7 = \frac{16}{27} \cdot \frac{-\frac{2187}{128} - 1}{-\frac{3}{2} - 1} \]

Calculate the denominator:
\[ -\frac{3}{2} - 1 = -\frac{3}{2} - \frac{2}{2} = -\frac{5}{2} \]

### Step 5: Simplify the numerator
Now simplify the numerator:
\[ -\frac{2187}{128} - 1 = -\frac{2187}{128} - \frac{128}{128} = -\frac{2187 + 128}{128} = -\frac{2315}{128} \]

### Step 6: Substitute back into the formula
Now substitute the simplified numerator and denominator back into the formula:
\[ S_7 = \frac{16}{27} \cdot \frac{-\frac{2315}{128}}{-\frac{5}{2}} = \frac{16}{27} \cdot \frac{2315}{128} \cdot \frac{2}{5} \]

### Step 7: Calculate the final value
Now calculate:
\[ S_7 = \frac{16 \cdot 2315 \cdot 2}{27 \cdot 128 \cdot 5} \]

Calculating the numerator:
\[ 16 \cdot 2315 \cdot 2 = 74240 \]

Calculating the denominator:
\[ 27 \cdot 128 \cdot 5 = 17280 \]

Thus,
\[ S_7 = \frac{74240}{17280} \]

### Step 8: Simplify the fraction
Now simplify \( \frac{74240}{17280} \):
\[ S_7 = \frac{463}{108} \]

### Final Result
The sum of the first 7 terms of the series is:
\[ \boxed{\frac{463}{108}} \]