Find the sum of `n` terms of series `(x+y) + (x^2+ xy +y^2)+(x^3+x^2y+xy^2+y^3)+..................`

Explanation

To find the sum of the first `n` terms of the series \((x+y) + (x^2 + xy + y^2) + (x^3 + x^2y + xy^2 + y^3) + \ldots\), we can follow these steps:

### Step 1: Identify the pattern in the series
The series can be rewritten as:
- The first term is \(x + y\)
- The second term is \(x^2 + xy + y^2\)
- The third term is \(x^3 + x^2y + xy^2 + y^3\)

We can see that the \(n\)-th term of the series can be expressed as:
\[ T_n = x^n + x^{n-1}y + x^{n-2}y^2 + \ldots + y^n \]
This is a polynomial of degree \(n\) in \(x\) and \(y\).

### Step 2: Recognize the pattern of the sum
The sum of the \(n\)-th term can be expressed using the formula for the sum of a geometric series. Each term can be viewed as a sum of powers of \(x\) and \(y\).

### Step 3: Use the formula for the sum of a geometric series
The sum of the \(n\)-th term can be derived as follows:
\[ T_n = \frac{x^{n+1} - y^{n+1}}{x - y} \]
This formula arises from the fact that the terms can be rearranged into a geometric series.

### Step 4: Sum the first \(n\) terms
To find the sum \(S_n\) of the first \(n\) terms, we can sum the individual terms:
\[ S_n = \sum_{k=1}^{n} T_k = \sum_{k=1}^{n} \frac{x^{k+1} - y^{k+1}}{x - y} \]
This can be simplified further.

### Step 5: Simplify the expression
We can factor out \(\frac{1}{x - y}\):
\[ S_n = \frac{1}{x - y} \sum_{k=1}^{n} (x^{k+1} - y^{k+1}) \]
This can be split into two separate sums:
\[ S_n = \frac{1}{x - y} \left( \sum_{k=1}^{n} x^{k+1} - \sum_{k=1}^{n} y^{k+1} \right) \]

### Step 6: Calculate the individual sums
Using the formula for the sum of a geometric series:
\[ \sum_{k=1}^{n} x^{k+1} = x^2 \frac{x^n - 1}{x - 1} \]
\[ \sum_{k=1}^{n} y^{k+1} = y^2 \frac{y^n - 1}{y - 1} \]

### Step 7: Combine the results
Substituting these back into the expression for \(S_n\):
\[ S_n = \frac{1}{x - y} \left( x^2 \frac{x^{n} - 1}{x - 1} - y^2 \frac{y^{n} - 1}{y - 1} \right) \]

### Final Result
Thus, the sum of the first \(n\) terms of the series is:
\[ S_n = \frac{x^2 (x^n - 1)}{(x - 1)(x - y)} - \frac{y^2 (y^n - 1)}{(y - 1)(x - y)} \]