The coefficient of x in the expansion of `(1+x)(1+2x)(1+3x)….(1+100x)` is :

Options

A. 5050
B. 10100
C. 5151
D. 4950

Answer

To find the coefficient of \( x \) in the expansion of \( (1+x)(1+2x)(1+3x)\cdots(1+100x) \), we can follow these steps:

### Step 1: Understand the Structure of the Expansion
The expression consists of 100 factors, each of the form \( (1+kx) \) where \( k \) ranges from 1 to 100. When we expand this product, we will obtain terms that are products of \( x \) raised to various powers.

### Step 2: Identify How to Get the Coefficient of \( x \)
To find the coefficient of \( x \), we need to select \( x \) from one of the factors and \( 1 \) from all the other factors. This means we will choose \( x \) from one factor \( (1+kx) \) and \( 1 \) from the remaining 99 factors.

### Step 3: Calculate the Contribution from Each Factor
If we choose \( x \) from the \( k \)-th factor \( (1+kx) \), the contribution to the coefficient of \( x \) will be \( k \). Thus, the total coefficient of \( x \) in the expansion will be the sum of all \( k \) from 1 to 100.

### Step 4: Sum the Values from 1 to 100
The coefficient of \( x \) is given by:
\[ 1 + 2 + 3 + \ldots + 100 \]
This is an arithmetic series where:
- The first term \( a = 1 \)
- The last term \( l = 100 \)
- The number of terms \( n = 100 \)

The sum \( S \) of the first \( n \) terms of an arithmetic series can be calculated using the formula:
\[ S = \frac{n}{2} \times (a + l) \]

### Step 5: Substitute the Values
Substituting the values we have:
\[ S = \frac{100}{2} \times (1 + 100) = 50 \times 101 = 5050 \]

### Conclusion
Thus, the coefficient of \( x \) in the expansion of \( (1+x)(1+2x)(1+3x)\cdots(1+100x) \) is \( 5050 \).