Ta có \(a^{14}+b^{14}=a^{15}+b^{15}\Leftrightarrow a^{15}-a^{14}=b^{14}-b^{15}\Leftrightarrow a^{14}\left(a-1\right)=b^{14}\left(1-b\right)\Leftrightarrow\dfrac{a-1}{1-b}=\dfrac{b^{14}}{a^{14}}\left(1\right)\)
ta lại có \(a^{15}+b^{15}=a^{16}+b^{16}\Leftrightarrow a^{16}-a^{15}=b^{15}-b^{16}\Leftrightarrow a^{15}\left(a-1\right)=b^{15}\left(1-b\right)\Leftrightarrow\dfrac{a-1}{b-1}=\dfrac{b^{15}}{a^{15}}\left(2\right)\)
Từ (1),(2)\(\Rightarrow\dfrac{b^{14}}{a^{14}}=\dfrac{b^{15}}{a^{15}}\Leftrightarrow\dfrac{b^{15}}{a^{15}}-\dfrac{b^{14}}{a^{14}}=0\Leftrightarrow\dfrac{b^{14}}{a^{14}}\left(\dfrac{a}{b}-1\right)=0\Leftrightarrow\dfrac{a}{b}-1=0\)(vì \(\dfrac{a^{14}}{b^{14}}\) là số dương)\(\Leftrightarrow\dfrac{a}{b}=1\Leftrightarrow a=b\)
Vậy thay vào P=2015a-2016b=2015a-2016a=-a=-b
Vậy P=-a=-b
