Tính giá trị các biểu thức sau:
a. A= (1+1/1.3).(1+1/2.4).(1+1/3.5).....(1+1/2014/2016)
b. B= (1/2-1).(1/3-1).(1/4-1).....(1/998-1).(1/999-1)
Để ý: [tex]a^2-1=a^2-a+a-1=a(a-1)+1(a-1)=(a-1)(a+1)[/tex]
Ta có:
[tex]A=(1+\frac{1}{1.3}).(1+\frac{1}{2.4}).(1+\frac{1}{3.5}).....(1+\frac{1}{2014.2016})\\=\frac{1.3+1}{1.3}.\frac{2.4+1}{2.4}.\frac{3.5+1}{3.5}.....\frac{2014.2016+1}{2014.2016}\\=\frac{2^2-1+1}{1.3}.\frac{3^2-1+1}{2.4}.\frac{4^2-1+1}{3.5}.....\frac{2015^2-1}{2014.2016}\\=\frac{2.2}{1.3}.\frac{3.3}{2.4}.\frac{4.4}{3.5}.....\frac{2015.2015}{2014.2016}\\=\frac{2.2.3.3.4.4.....2015.2015}{1.3.2.4.3.5.....2014.2016}\\=\frac{ (2.3.4.....2015).(2.3.4.....2015)}{(1.2.3.....2014).(3.4.5.....2016)}\\=\frac{(2.3.4.....2015)}{(1.2.3.....2014)}.\frac{(2.3.4.....2015)}{(3.4.5.....2016)}\\=\frac{2015}{1}.\frac{2}{2016}\\=\frac{2015}{1}.\frac{1}{1008}\\=\frac{2015}{1008}[/tex]
[tex]B=(\frac{1}{2}-1).(\frac{1}{3}-1).(\frac{1}{4}-1).....(\frac{1}{999}-1)\\=\frac{-1}{2}.\frac{-2}{3}.\frac{-3}{4}.....\frac{-998}{999}[/tex]
B có số thừa số là: [tex]999-2+1=998[/tex] số
[tex]\Rightarrow B=\frac{(-1)(-2)(-3)...(-998)}{2.3.4.....999}\\=\frac{(-1).(-1).(-1)...(-1)}{999}(998so-1)\\=\frac{(-1)^{998}}{999}\\=\frac{1}{999}[/tex]