Not as simple as it seems.

Probability Level 5

Let P 0 ( x ) = x 3 + 313 x 2 − 77 x − 8 P_0(x) = x^3 + 313x^2 - 77x - 8 , For integers n ≥ 1 n \ge 1 ,, define P n ( x ) = P n − 1 ( x − n ) P_n(x) = P_{n - 1}(x - n) , What is the coefficient of x x , in P 20 ( x ) P_{20}(x) ?


The answer is 763.

Parth Lohomi by Parth Lohomi , 20, India

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1 solution

It follows from the definition of P n P_n that P n ( x ) = P 0 ( x − n ( n + 1 ) 2 ) P_n(x) = P_0\left(x - \frac{n(n+1)}{2}\right) (Proof by induction)

Hence, P 20 ( x ) = P 0 ( x − 210 ) = ( x − 210 ) 3 + 313 ( x − 210 ) 2 − 77 ( x − 210 ) − 8 P_{20}(x) = P_{0}(x - 210) = (x - 210)^3 + 313 (x - 210)^2 - 77 (x - 210) - 8 and therefore the coefficient of x x must be 3 ⋅ 21 0 2 − 313 ⋅ 2 ⋅ 210 − 77 = 763 3 \cdot 210^2 - 313 \cdot 2 \cdot 210 - 77 = \boxed{763}

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