Question

Fie P un polinom cu coeficientii reali astfel incat
$p(1) + p(2) + ... + p(n) = n ^{5}$
pentru orice numar natural
$n \geqslant 1$
Sa se calculeze
$p( \frac{3}{2} )$


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Answer 1

$\begin{array}{lcl}P(1)+P(2)+...+P(n)&=& n^5\\P(1)+P(2)+...+P(n)+P(n+1) &=& (n+1)^5 \end{array}\\ \\\\ \text{Scadem cele 2 relatii:}\\ \\ \Rightarrow P(n+1) = (n+1)^5 - n^5\\ \\ \Rightarrow P(n) = n^5-(n-1)^5,\quad n\in \mathbb{N}^* \\ \\ \Rightarrow P(x) = x^5 - (x-1)^5,\quad x\in \mathbb{R} \\ \\\Rightarrow P\Big(\dfrac{3}{2}\Big) =\Big(\dfrac{3}{2}\Big)^5 - \Big(\dfrac{3}{2}-1\Big)^5 \\ \\ \Rightarrow P\Big(\dfrac{3}{2}\Big) = \dfrac{3^5 - 1^5}{2^5}\\ \\ \Rightarrow P\Big(\dfrac{3}{2}\Big) = \dfrac{242}{32}$

$\Rightarrow \boxed{P\Big(\dfrac{3}{2}\Big) = \dfrac{121}{16}}$