Find the sum of n terms of the series 4/3 + 10/9+28/27+…

Explanation

To find the sum of the first n terms of the series \( S = \frac{4}{3} + \frac{10}{9} + \frac{28}{27} + \ldots \), we can follow these steps:

### Step 1: Rewrite the terms of the series
We can express each term in the series in a more manageable form:

\[ \frac{4}{3} = 1 + \frac{1}{3}, \quad \frac{10}{9} = 1 + \frac{1}{9}, \quad \frac{28}{27} = 1 + \frac{1}{27} \]

Thus, we can rewrite the series as:

\[ S = \left(1 + \frac{1}{3}\right) + \left(1 + \frac{1}{9}\right) + \left(1 + \frac{1}{27}\right) + \ldots \]

### Step 2: Separate the constant and variable parts
Now, we can separate the constant part and the variable part:

\[ S = n + \left(\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots\right) \]

Here, \( n \) comes from the constant \( 1 \) in each term, summed over \( n \) terms.

### Step 3: Identify the geometric series
The remaining series \( \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots \) is a geometric series where:

- First term \( a = \frac{1}{3} \)
- Common ratio \( r = \frac{1}{3} \)

### Step 4: Use the formula for the sum of a geometric series
The sum of the first \( n \) terms of a geometric series can be calculated using the formula:

\[ S_n = a \frac{1 - r^n}{1 - r} \]

Substituting the values of \( a \) and \( r \):

\[ \text{Sum of the series} = \frac{1}{3} \cdot \frac{1 - \left(\frac{1}{3}\right)^n}{1 - \frac{1}{3}} = \frac{1}{3} \cdot \frac{1 - \frac{1}{3^n}}{\frac{2}{3}} = \frac{1}{2} \left(1 - \frac{1}{3^n}\right) \]

### Step 5: Combine the results
Now, we can combine the results from steps 2 and 4:

\[ S = n + \frac{1}{2} \left(1 - \frac{1}{3^n}\right) \]

### Step 6: Final expression for the sum
Thus, the sum of the first \( n \) terms of the series is:

\[ S = n + \frac{1}{2} - \frac{1}{2 \cdot 3^n} \]

### Summary
The final result for the sum of the first \( n \) terms of the series \( \frac{4}{3} + \frac{10}{9} + \frac{28}{27} + \ldots \) is:

\[ S = n + \frac{1}{2} - \frac{1}{2 \cdot 3^n} \]

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