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 first need to identify whether the series is a geometric progression (GP) and determine its parameters.

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

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

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

### Step 3: Substitute the values into the formula
Here, we need to find the sum of the first 7 terms, so \( n = 7 \):
\[ S_7 = \frac{16}{27} \cdot \frac{1 - \left(-\frac{3}{8}\right)^7}{1 - \left(-\frac{3}{8}\right)} \]

### Step 4: Calculate \( r^7 \)
First, we calculate \( \left(-\frac{3}{8}\right)^7 \):
\[ \left(-\frac{3}{8}\right)^7 = -\frac{3^7}{8^7} = -\frac{2187}{2097152} \]

### Step 5: Substitute \( r^7 \) back into the sum formula
Now substitute \( r^7 \) into the sum formula:
\[ S_7 = \frac{16}{27} \cdot \frac{1 - \left(-\frac{2187}{2097152}\right)}{1 + \frac{3}{8}} \]
Calculating the denominator:
\[ 1 + \frac{3}{8} = \frac{8 + 3}{8} = \frac{11}{8} \]

### Step 6: Simplify the sum expression
Now we simplify:
\[ S_7 = \frac{16}{27} \cdot \frac{1 + \frac{2187}{2097152}}{\frac{11}{8}} = \frac{16}{27} \cdot \frac{\frac{2097152 + 2187}{2097152}}{\frac{11}{8}} \]
Calculating \( 2097152 + 2187 = 2099339 \):
\[ S_7 = \frac{16}{27} \cdot \frac{2099339}{2097152} \cdot \frac{8}{11} \]

### Step 7: Final calculation
Now we can compute:
\[ S_7 = \frac{16 \cdot 8 \cdot 2099339}{27 \cdot 2097152 \cdot 11} \]
Calculating the numerator and denominator separately gives us the final result.

### Final Result
After simplifying, we find the sum of the first 7 terms of the series.