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

Explanation

To find the sum of the first n terms of the series \((x+y)+(x^2+xy+y^2)+(x^3+x^2y+xy^2+y^3)+...\), 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 \(x^3 + x^2y + xy^2 + y^3\)

Each term can be seen as the expansion of \((x+y)^n\) for \(n = 1, 2, 3, \ldots\).

### Step 2: Write the general term
The \(n\)-th term of the series can be expressed as:
\[ T_n = (x+y)^n \]

### Step 3: Write the sum of the first n terms
The sum of the first n terms \(S_n\) can be expressed as:
\[ S_n = (x+y)^1 + (x+y)^2 + (x+y)^3 + \ldots + (x+y)^n \]

This is a geometric series where:
- The first term \(a = (x+y)\)
- The common ratio \(r = (x+y)\)
- The number of terms \(n\)

### Step 4: Use the formula for the sum of a geometric series
The sum of the first n terms of a geometric series is given by:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
Substituting the values we have:
\[ S_n = (x+y) \frac{1 - (x+y)^n}{1 - (x+y)} \]

### Step 5: Simplify the expression
Thus, the sum of the first n terms can be simplified to:
\[ S_n = \frac{(x+y)(1 - (x+y)^n)}{1 - (x+y)} \]

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