Question

[Nguyên Minh Hiếu]

Cho A =1/2x3/4x5/6x......x2011/2012 . Chứng minh  A²<1/2013

Answer 1

[Akai Haruma]

Lời giải:

$A=\frac{1.3.5....2011}{2.4.6....2012}$
$A^2=\frac{1.3}{2^2}.\frac{3.5}{4^2}.\frac{5.7}{6^2}....\frac{2009.2011}{2010^2}.\frac{2011}{2012^2}$

$=\frac{3}{4}.\frac{15}{16}.\frac{35}{36}....\frac{4040099}{4040100}.\frac{2011}{2012^2}$

$< 1.1.1.....1.\frac{2011}{2012^2}=\frac{2011}{2012^2}$

$<\frac{2011}{2012^2-1}=\frac{2011}{2011.2013}=\frac{1}{2013}$

Ta có đpcm.