In the expression `(x+1) (x+4)(x+9)(x+16)"….."(x+400)`, the coefficient of `x^(19)` is

Options

A. 2870
B. 2100
C. 4001
D. 1900

Answer

To find the coefficient of \( x^{19} \) in the expression \( (x+1)(x+4)(x+9)(x+16)\ldots(x+400) \), we can follow these steps:

### Step 1: Identify the terms in the expression
The expression consists of 20 factors:
- \( (x + 1^2) \)
- \( (x + 2^2) \)
- \( (x + 3^2) \)
- ...
- \( (x + 20^2) \)

This means we have the terms \( (x + 1)(x + 4)(x + 9)(x + 16)\ldots(x + 400) \), where the constants are the squares of the integers from 1 to 20.

### Step 2: Determine how to obtain \( x^{19} \)
To get \( x^{19} \), we need to take \( x \) from 19 of the 20 brackets and the constant from the remaining bracket.

### Step 3: Calculate the contribution from each case
For each case where we take \( x \) from 19 brackets and the constant from the 20th bracket, the constant will be \( k^2 \) where \( k \) is the index of the bracket we are not taking \( x \) from. Thus, the contribution to the coefficient of \( x^{19} \) will be the sum of the squares of the integers from 1 to 20.

### Step 4: Write the sum of squares formula
The sum of the squares of the first \( n \) natural numbers is given by the formula:
\[ \text{Sum} = \frac{n(n + 1)(2n + 1)}{6} \]
For \( n = 20 \):
\[ \text{Sum} = \frac{20(20 + 1)(2 \times 20 + 1)}{6} \]
\[ = \frac{20 \times 21 \times 41}{6} \]

### Step 5: Calculate the sum
Calculating the above expression:
\[ = \frac{20 \times 21 \times 41}{6} \]
\[ = \frac{20 \times 21 \times 41}{6} = \frac{20 \times 21 \times 41}{6} = \frac{20 \times 21 \times 41}{6} \]
Calculating \( 20 \times 21 = 420 \)
\[ = \frac{420 \times 41}{6} = \frac{17220}{6} = 2870 \]

### Final Answer
Thus, the coefficient of \( x^{19} \) in the expression is **2870**.

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