Tính: \(\left(1+\frac{1}{3}\right)\left(1+\frac{1}{8}\right)\left(1+\frac{1}{15}\right).....\left(1+\frac{1}{120}\right)\) =...
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16 tháng 2 2017
\(\left(1+\frac{1}{3}\right)\left(1+\frac{1}{8}\right)\left(1+\frac{1}{15}\right)...\left(1+\frac{1}{120}\right)\)
\(=\frac{4}{3}.\frac{9}{8}.\frac{16}{15}...\frac{121}{120}\)
\(=\frac{2.2}{1.3}.\frac{3.3}{2.4}.\frac{4.4}{3.5}...\frac{11.11}{10.12}\)
\(=\frac{2.2.3.3.4.4...11.11}{1.3.2.4.3.5...10.12}\)
\(=\frac{\left(2.3.4...11\right)\left(2.3.4...11\right)}{\left(1.2.3...10\right).\left(3.2.5...12\right)}\)
\(=\frac{11.2}{1.12}\)
\(=\frac{11}{6}\)
18 tháng 5 2017
\(\left(1-\frac{1}{3}\right).\left(1-\frac{1}{5}\right).\left(1-\frac{1}{7}\right)...\left(1-\frac{1}{2}\right).\left(1-\frac{1}{4}\right).\left(1-\frac{1}{6}\right).\left(1-\frac{1}{120}\right)\)
\(=\frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{1}{2}.\frac{3}{4}.\frac{5}{6}.\frac{119}{120}=\left(\frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{118}{119}\right).\left(\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{119}{120}\right)\)
\(=\frac{\left(2.4.6...118\right).\left(1.3.5...119\right)}{\left(3.5.7...119\right).\left(2.4.6...120\right)}=\frac{1}{120}\).
Đáp số: \(\frac{1}{120}\).
3 tháng 6 2016
A = ( 1/3 - 1).( 1/6 - 1).( 1/10 - 1).( 1/15 - 1).( 1/21 - 1).( 1/28 - 1).( 1/36 - 1)
A = -2/3 . ( -5/6) . ( -9/10) . ( -14/15) . ( -20/21) . ( -27/28) . ( -35/36)
Do tích A có lẻ thừa số, mỗi thừa số đều mang dấu âm nên A mang dấu âm
A = -[ 2/3 . 5/6 . 9/10 . 14/15 . 20/21 . 27/28 . 35/36]
Đặt B = 2/3 . 5/6 . 9/10 . 14/15 . 20/21 . 27/28 . 35/36
B = 4/6 . 10/12 . 18/20 . 28/30 . 40/42 . 54/56 . 70/72
B = 1.4/2.3 . 2.5/3.4 . 3.6/4.5 . 4.7/5.6 . 5.8/6.7 . 6.9/7.8 . 7.10/8.9
B = 1.2.3.4.5.6.7/3.4.5.6.7.8.9 . 4.5.6.7.8.9.10/2.3.4.5.6.7.8
B = 2/8.9 . 9.10/2.3
B = 5/12
A = -5/12
\(\left(1+\dfrac{1}{3}\right)\left(1+\dfrac{1}{8}\right)\left(1+\dfrac{1}{15}\right)...\left(1+\dfrac{1}{120}\right)\)
\(=\dfrac{4}{3}.\dfrac{9}{8}.\dfrac{16}{15}...\dfrac{121}{120}=\dfrac{2.2}{1.3}.\dfrac{3.3}{2.4}.\dfrac{4.4}{3.5}.\dfrac{11.11}{10.12}\)
\(=\dfrac{2}{1}.\dfrac{11}{12}=\dfrac{11}{6}\)