To find the coefficient of \( x^{18} \) in the expression \( (x^2 + 1)(x^2 + 4)(x^2 + 9) \cdots (x^2 + 100) \), we can follow these steps:
### Step 1: Understand the Expression
The expression consists of factors of the form \( (x^2 + k^2) \) where \( k \) ranges from 1 to 10. Therefore, we can rewrite the expression as:
\[
(x^2 + 1^2)(x^2 + 2^2)(x^2 + 3^2) \cdots (x^2 + 10^2)
\]
### Step 2: Identify the Required Power
We need to find the coefficient of \( x^{18} \). Since each \( (x^2 + k^2) \) contributes either \( x^2 \) or \( k^2 \), we need to select terms such that the total power of \( x \) sums to 18.
### Step 3: Determine the Number of \( x^2 \) Terms
To achieve a total power of \( x^{18} \), we can select \( x^2 \) from 9 of the factors and the constant term from 1 factor. This is because:
\[
2n = 18 \implies n = 9
\]
This means we choose \( x^2 \) from 9 factors and the constant term (which is \( k^2 \)) from 1 factor.
### Step 4: Choose the Constant Term
The constant term can be chosen from any one of the 10 factors. The possible constant terms are \( 1^2, 2^2, 3^2, \ldots, 10^2 \), which are \( 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 \).
### Step 5: Calculate the Coefficient
The coefficient of \( x^{18} \) will be the sum of the squares of the numbers from 1 to 10, as each square represents the constant term chosen from one factor while the remaining factors contribute \( x^2 \).
The sum of squares formula is given by:
\[
\text{Sum of squares} = \frac{n(n + 1)(2n + 1)}{6}
\]
where \( n = 10 \).
### Step 6: Substitute and Calculate
Substituting \( n = 10 \):
\[
\text{Sum of squares} = \frac{10(10 + 1)(2 \cdot 10 + 1)}{6} = \frac{10 \cdot 11 \cdot 21}{6}
\]
### Step 7: Simplify
Calculating this:
\[
= \frac{10 \cdot 11 \cdot 21}{6} = \frac{2310}{6} = 385
\]
### Final Result
Thus, the coefficient of \( x^{18} \) in the given expression is:
\[
\boxed{385}
\]
