b: \(12\left(x-2015\right)^2=36-y^2\)
=>\(36-y^2\) ⋮12 và \(36-y^2\ge0\)
=>\(y^2\) ⋮12 và \(y^2\le36\)
=>\(y^2\) ⋮12 và 0<=y<=6
=>y∈{0;6}
TH1: y=0
\(12\left(x-2015\right)^2=36-y^2\)
=>\(12\left(x-2015\right)^2=36-0=36\)
=>\(\left(x-2015\right)^2=3\) (vô lý vì x là số tự nhiên)
TH2: y=6
\(12\left(x-2015\right)^2=36-y^2\)
=>\(12\left(x-2015\right)^2=36-36=0\)
=>\(\left(x-2015\right)^2=0\)
=>x-2015=0
=>x=2015
Bài 1:
a: \(9^9\cdot5\cdot4^{15}-8^9\cdot4\cdot3^{20}\)
\(=3^{18}\cdot5\cdot2^{30}-2^{27}\cdot2^2\cdot3^{20}\)
\(=5\cdot3^{18}\cdot2^{30}-2^{29}\cdot3^{20}=2^{29}\cdot3^{18}\left(5\cdot2-3^2\right)=2^{29}\cdot3^{18}\)
Ta có: \(6^{19}\cdot5\cdot2^9-27^6\cdot7\cdot2^{29}\)
\(=2^{19}\cdot3^{19}\cdot5\cdot2^9-3^{18}\cdot7\cdot2^{29}\)
\(=2^{28}\cdot3^{19}\cdot5-2^{29}\cdot3^{18}\cdot7=2^{28}\cdot3^{18}\left(3\cdot5-2\cdot7\right)=2^{28}\cdot3^{18}\)
Ta có: \(A=\frac{9^9\cdot5\cdot4^{15}-8^9\cdot4\cdot3^{20}}{6^{19}\cdot5\cdot2^9-27^6\cdot7\cdot2^{29}}\)
\(=\frac{2^{29}\cdot3^{18}}{2^{28}\cdot3^{18}}=2\)
b: \(\frac{1}{1\cdot2001}+\frac{1}{2\cdot2002}+\cdots+\frac{1}{15\cdot2015}\)
\(=\frac{1}{2000}\left(\frac{2000}{1\cdot2001}+\frac{2000}{2\cdot2002}+\cdots+\frac{2000}{15\cdot2015}\right)\)
\(=\frac{1}{2000}\left(1+\frac12+\cdots+\frac{1}{15}-\frac{1}{2001}-\frac{1}{2002}-\cdots-\frac{1}{2015}\right)\)
Ta có: \(\frac{1}{1\cdot16}+\frac{1}{2\cdot17}+\cdots+\frac{1}{2000\cdot2015}\)
\(=\frac{1}{15}\left(\frac{15}{1\cdot16}+\frac{15}{2\cdot17}+\cdots+\frac{15}{2000\cdot2015}\right)\)
\(=\frac{1}{15}\left(1-\frac{1}{16}+\frac12-\frac{1}{17}+\cdots+\frac{1}{2000}-\frac{1}{2015}\right)\)
\(=\frac{1}{15}\left(1+\frac12+\cdots+\frac{1}{2000}-\frac{1}{16}-\frac{1}{17}-\cdots-\frac{1}{2015}\right)\)
\(=\frac{1}{15}\left(1+\frac12+\cdots+\frac{1}{15}-\frac{1}{2001}-\frac{1}{2002}-\cdots-\frac{1}{2015}\right)\)
Ta có: \(B=\frac{\frac{1}{1\cdot2001}+\frac{1}{2\cdot2002}+\cdots+\frac{1}{15\cdot2015}}{\frac{1}{1\cdot16}+\frac{1}{2\cdot17}+\cdots+\frac{1}{2000\cdot2015}}\)
\(=\frac{1}{2000}:\frac{1}{15}=\frac{15}{2000}=\frac{3}{400}\)
