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\(\left(-2\right).\left(-1\frac{1}{2}\right).\left(-1\frac{1}{3}\right)...\left(-1\frac{1}{2013}\right)\)
\(=\frac{-2}{1}.\frac{-3}{2}.\frac{-4}{3}...\frac{-2014}{2013}\)
\(=-\left(\frac{2}{1}.\frac{3}{2}.\frac{4}{3}...\frac{2014}{2013}\right)\)(vì có lẻ thừa số âm)
\(=-2014\)
\(\left(\frac{1}{2}-1\right)\left(\frac{1}{3}-1\right)\left(\frac{1}{4}-1\right)...\left(\frac{1}{2012}-1\right)\left(\frac{1}{2013}-1\right)\)
\(=\frac{-1}{2}.\frac{-2}{3}.\frac{-3}{4}...\frac{-2011}{2012}.\frac{-2012}{2013}\)
\(=\frac{\left(-1\right).\left(-2\right).\left(-3\right)...\left(-2011\right).\left(-2012\right)}{2.3.4....2013}\)
\(=\frac{1.2.3...2011.2012}{2.3.4.5...2013}\) ( vì các số hạng ở trên tử chẵn )
\(=\frac{1}{2013}\)
\(\left(\frac{1}{2}-1\right)\left(\frac{1}{3}-1\right)\left(\frac{1}{4}-1\right)...\left(\frac{1}{2012}-1\right)\left(\frac{1}{2013}-1\right)\)
\(=\frac{-1}{2}.\frac{-2}{3}.\frac{-3}{4}...\frac{-2011}{2012}.\frac{-2012}{2013}\)
\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{2011}{2012}.\frac{2012}{2013}\)(vì có 2012 thừa số âm nên kết quả là dương)
\(=\frac{1}{2013}\)
\(\left(-2\right).\left(-1\frac{1}{2}\right)\left(-1\frac{1}{3}\right).....\left(-1\frac{1}{2013}\right)\)
\(=\left(-2\right).\left(\frac{-3}{2}\right)\left(-\frac{4}{3}\right)......\left(\frac{-2014}{2013}\right)\)
\(=\frac{\left(-2\right).\left(-3\right).\left(-4\right)....\left(-2014\right)}{2.3.....2013}\)
\(=\frac{2.3.4....2014\left(\text{Vì có 2014 thừa số âm }\right)}{2.3....2013}\)
\(=\frac{\left(2.3.4....2013\right).2014}{2.3....2013}\)
\(=2014\)
\(\left(-2\right)\left(-1\frac{1}{2}\right)...........\left(-1\frac{1}{2010}\right)=\frac{\left[\left(-2\right)\left(-3\right).....\left(-2010\right)\right].\left(-2011\right)}{\left(2.3.4.............2010\right)}=\frac{\left(-1\right)\left(-2011\right)}{1}=2011\)
\(\left(-2\right).\left(-1\frac{1}{2}\right).\left(-1\frac{1}{3}\right)....\left(-1\frac{1}{2009}\right).\left(-1\frac{1}{2010}\right)=\left(-2\right).\left(-\frac{3}{2}\right).\left(-\frac{4}{3}\right)....\left(-\frac{2010}{2009}\right).\left(-\frac{2011}{2010}\right)=\frac{\left(-2\right).\left(-3\right).\left(-4\right)....\left(-2010\right).\left(-2011\right)}{2.3.4....2009.2010}\)=2011
Ta áp dụng công thức: \(a-b=\left[-\left(b-a\right)\right]\)
\(\left(\frac{1}{2}-1\right)\left(\frac{1}{3}-1\right)\left(\frac{1}{4}-1\right)...\left(\frac{1}{2012}-1\right)\left(\frac{1}{2013}-1\right)\)
\(=-\left[\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)...\left(1-\frac{1}{2012}\right)\left(1-\frac{1}{2013}\right)\right]\)
\(=-\left(\frac{1}{2}.\frac{2}{3}.\frac{3}{4}....\frac{2011}{2012}.\frac{2012}{2013}\right)\)
\(=-\frac{1.2.3...2011.2012}{2.3.4....2012.2013}\)
\(=-\frac{1}{2013}\)
\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}......\frac{2012}{2013}\)
Liệt tử thừa với mẫu thừa:
\(=\frac{1}{2013}\)
Chúc em học tốt^^