[tex]\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 \\ \\ \Rightarrow P(x) = x^5 - (x-1)^5 \\ \\\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}[/tex]
[tex]\Rightarrow \boxed{P\Big(\dfrac{3}{2}\Big) = \dfrac{121}{16}}[/tex]
