# In the polynomial (x-1)(x-2)(x-3)cdots(x-100). Find the coefficient of x^99 is?

Oct 23, 2017

${a}_{99} = 5050$

#### Explanation:

Calling ${p}_{n} \left(x\right) = {\prod}_{k = 1}^{n} \left(x - k\right)$ we have

we now that their roots are $1 , 2 , 3 , \cdots , n$

and the coefficients for

${p}_{n} \left(x\right) = {a}_{0} + {a}_{1} x + {a}_{2} {x}^{2} + \cdots + {a}_{n - 1} {x}^{n - 1} + {x}^{n}$

are liked to the roots values - For instance

a_0 = n!, a_(n-1) = sum_(k=1)^n k = (n(n+1))/2

so ${a}_{99} = \frac{100 \left(100 + 1\right)}{2} = 5050$