If `(1+px+x^(2))^(n)=1+a_(1)x+a_(2)x^(2)+…+a_(2n)x^(2n)`.
Which of the following is true for `1 lt r lt 2n`

Options

A. (a) `(np+pr)a_(r )=(r+1)a_(r+1)+(r-1)a_(r-1)`
B. (b) `(np-pr)a_(r )=(r+1)a_(r+1)+(r-1-2n)a_(r-1)`
C. (c) `(np-pr)a_(r )=(r+1)a_(r+1)+(r-1-n)a_(r-1)`
D. (d) `(2np+pr)a_(r )=(r+1+n)a_(r+1)+(r+1-n)a_(r-1)`

Explanation

To solve the problem, we start with the expression given in the question:

\[ (1 + px + x^2)^n = 1 + a_1 x + a_2 x^2 + \ldots + a_{2n} x^{2n} \]

We need to analyze the coefficients \( a_r \) for \( 1 < r < 2n \).

### Step 1: Differentiate both sides with respect to \( x \)

Differentiating the left-hand side using the chain rule:

\[ \frac{d}{dx}[(1 + px + x^2)^n] = n(1 + px + x^2)^{n-1} \cdot (p + 2x) \]

For the right-hand side, we differentiate term by term:

\[ \frac{d}{dx}[1 + a_1 x + a_2 x^2 + \ldots + a_{2n} x^{2n}] = a_1 + 2a_2 x + 3a_3 x^2 + \ldots + 2n a_{2n} x^{2n-1} \]

### Step 2: Set the derivatives equal to each other

Now we have:

\[ n(1 + px + x^2)^{n-1} \cdot (p + 2x) = a_1 + 2a_2 x + 3a_3 x^2 + \ldots + 2n a_{2n} x^{2n-1} \]

### Step 3: Multiply both sides by \( (1 + px + x^2) \)

Next, we multiply both sides by \( (1 + px + x^2) \):

\[ n(1 + px + x^2)^{n} \cdot (p + 2x) = (a_1 + 2a_2 x + \ldots + 2n a_{2n} x^{2n-1})(1 + px + x^2) \]

### Step 4: Compare coefficients of \( x^r \)

We need to find the coefficient of \( x^r \) on both sides.

On the left-hand side, the coefficient of \( x^r \) can be derived from:

- The term \( n \cdot p \) from the differentiation.
- The contributions from the expansion of \( (1 + px + x^2)^n \).

On the right-hand side, we will have contributions from:

- \( a_r \)
- \( p \cdot r \cdot a_r \)
- \( (r-1) \cdot a_{r-1} \)
- Other terms depending on the coefficients of \( x^{r-1} \) and \( x^{r-2} \).

### Step 5: Formulate the equation

Setting the coefficients equal gives us:

\[ n p a_r + 2n a_{r-1} = (r + 1) a_{r + 1} + p \cdot r \cdot a_r + (r - 1) a_{r - 1} \]

### Step 6: Rearranging the equation

Rearranging the terms leads to:

\[ n p a_r - p \cdot r \cdot a_r = (r + 1) a_{r + 1} + (r - 1) a_{r - 1} - 2n a_{r - 1} \]

This can be simplified to:

\[ (n p - p r) a_r = (r + 1) a_{r + 1} + (r - 1 - 2n) a_{r - 1} \]

### Conclusion

From this equation, we can analyze the behavior of the coefficients \( a_r \) for \( 1 < r < 2n \) and determine which among the given options is true.