In sub - part (i) to (x) choose the correct option and in sub - part (xi) to (xy), answer the questions as intructed .
The derivative of `1+x+x^(2)+x(3)+….+x^(50)` at x=1:

Options

A. 1275
B. 1725
C. 1257
D. 1255

Answer

To find the derivative of the function \( f(x) = 1 + x + x^2 + x^3 + \ldots + x^{50} \) at \( x = 1 \), we can follow these steps:

### Step 1: Write the function
The function can be expressed as:
\[ f(x) = 1 + x + x^2 + x^3 + \ldots + x^{50} \]

### Step 2: Find the derivative
To find the derivative \( f'(x) \), we differentiate each term of the function:
- The derivative of \( 1 \) is \( 0 \).
- The derivative of \( x \) is \( 1 \).
- The derivative of \( x^2 \) is \( 2x \).
- The derivative of \( x^3 \) is \( 3x^2 \).
- Continuing this pattern, the derivative of \( x^n \) is \( nx^{n-1} \).

Thus, the derivative \( f'(x) \) is:
\[ f'(x) = 0 + 1 + 2x + 3x^2 + 4x^3 + \ldots + 50x^{49} \]

### Step 3: Evaluate the derivative at \( x = 1 \)
Now, we substitute \( x = 1 \) into the derivative:
\[ f'(1) = 1 + 2(1) + 3(1^2) + 4(1^3) + \ldots + 50(1^{49}) \]
This simplifies to:
\[ f'(1) = 1 + 2 + 3 + 4 + \ldots + 50 \]

### Step 4: Calculate the sum
The sum of the first \( n \) natural numbers can be calculated using the formula:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
For \( n = 50 \):
\[ \text{Sum} = \frac{50(50 + 1)}{2} = \frac{50 \times 51}{2} = 1275 \]

### Conclusion
Thus, the derivative of \( 1 + x + x^2 + x^3 + \ldots + x^{50} \) at \( x = 1 \) is:
\[ \boxed{1275} \]