\(60!=1\cdot2\cdot3\cdot4\cdot5\cdot...\cdot59\cdot60=1\cdot3\cdot5\cdot...\cdot57\cdot59\times2\cdot4\cdot6\cdot...\cdot58\cdot60\)

\(=1\cdot3\cdot5\cdot...\cdot57\cdot59\times2^{30}\cdot1\cdot2\cdot3\cdot...\cdot30=1\cdot3\cdot5\cdot...\cdot57\cdot59\times2^{30}\times30!\)

\(\Rightarrow1\cdot3\cdot5\cdot...\cdot59=\frac{60!}{30!\times2^{30}}=\frac{31}{2}\cdot\frac{32}{2}\cdot\frac{33}{2}\cdot...\cdot\frac{60}{2}\)đpcm.

\(\frac{31}{2}\cdot\frac{32}{2}\cdot...\cdot\frac{60}{2}\cdot2\cdot4\cdot...\cdot58\cdot60\)

=31.32.33.34...60.1.2.3.4.5...29.30

=1.2.3.4.5.6.7.8.9.10...60

1.3.5.7...59.2.4.6.8...60

=1.2.3.4.5.6...60

Vậy \(\frac{31}{2}\cdot\frac{32}{2}\cdot\frac{33}{2}\cdot...\cdot\frac{60}{2}=1\cdot3\cdot5\cdot...\cdot59\)

Ta có: \(\frac{31}{2}.\frac{32}{2}...\frac{60}{2}=\frac{31.32...60}{2^{30}}=31.33...57.59.\left(\frac{32.34...58.60}{2^{30}}\right)\)

                                                                         \(=31.33...57.59.\left(\frac{16.17...29.30}{2^{15}}\right)=17.19...27.29.31.33...57.59.\left(\frac{16.18...30}{2^{15}}\right)\)

\(=17.19...57.59.\left(\frac{8.9...15}{2^7}\right)=9.11.13.15.17...57.59.\left(\frac{8.10.12.14}{2^7}\right)\)

\(=9.11...57.59.\left(\frac{4.5.6.7}{2^3}\right)=5.7.9...57.59.\left(\frac{4.6}{2^3}\right)\)

\(=5.7.9...57.59.3=1.3.5...59\)

\(1.3.5....59=\frac{\left(1.3.5....59\right).\left(2.4.6....60\right)}{2.4.6....60}=\frac{\left(1.2.3.4.5...30\right).31....59.60}{2^{30}.\left(1.2.3...30\right)}=\frac{31.32....60}{2^{30}}=\frac{31}{2}.\frac{32}{2}.\frac{33}{2}...\frac{60}{2}\)

Chị quản lý Sao làm tốt thế mà chẳng được olm công nhận nhỉ

Vì đó là người quản lí nên công nhận cũng có được gì !!!

Ta có: \(\frac{31}{2}.\frac{32}{2}.\frac{33}{2}...\frac{60}{2}=\frac{31.32.33...60}{2.2.2...2}=\frac{31.32.33...60}{2^{30}}\)

                                                       (30 số 2)

\(=\frac{\left(31.32.33...60\right).\left(1.2.3...30\right)}{2^{30}.\left(1.2.3...30\right)}=\frac{31.32.33...60.1.2.3...30}{\left(2.1\right).\left(2.2\right).\left(2.3\right)...\left(2.30\right)}=\frac{\left(1.3.5...59\right).\left(2.4.6...60\right)}{\left(2.4.6...60\right)}=1.3.5...59\)

\(\frac{31}{2}.\frac{32}{2}.\frac{33}{2}...\frac{60}{2}=\frac{31.32.33...60}{2^{30}}\)

\(=\frac{\left(31.32.33...60\right).\left(1.2.3...30\right)}{2^{30}.\left(1.2.3...30\right)}\)

\(=\frac{1.2.3...60}{2^{30}\left(1.2.3...30\right)}\)

\(=\frac{\left(1.3.5.7...59\right)\left(2.4.6.8...60\right)}{\left(2.4.6.8...60\right)}\)

\(=1.3.5.7...59\)

Bài 1 :

\(\left(-2\right)\left(x+1\right)-3\left(1-x\right)=4\)

\(\Leftrightarrow-2x-2-3+3x=4\)

\(\Leftrightarrow x=4+2+3=9\)

Bài 2 :

Cho \(S=\frac{1}{31}+\frac{1}{32}+...+\frac{1}{60}\)

\(\Leftrightarrow S=\left(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40}\right)+\left(\frac{1}{41}+\frac{1}{42}+...+\frac{1}{50}\right)\)

\(+\left(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{60}\right)\)

\(\Rightarrow S< \left(\frac{1}{30}+\frac{1}{30}+...+\frac{1}{30}\right)+\left(\frac{1}{40}+\frac{1}{40}+...+\frac{1}{40}\right)\)

\(+\left(\frac{1}{50}+\frac{1}{50}+...+\frac{1}{50}\right)\)

\(\Leftrightarrow S< \frac{10}{30}+\frac{10}{40}+\frac{10}{50}=\frac{47}{60}< \frac{48}{60}=\frac{4}{5}\)⁽¹⁾

Lại có :

\(S=\left(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40}\right)+\left(\frac{1}{41}+\frac{1}{42}+...+\frac{1}{50}\right)\)

\(+\left(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{60}\right)\)

\(\Leftrightarrow S>\left(\frac{1}{40}+\frac{1}{40}+...+\frac{1}{40}\right)+\left(\frac{1}{50}+\frac{1}{50}+...+\frac{1}{50}\right)\)

\(+\left(\frac{1}{60}+\frac{1}{60}+...+\frac{1}{60}\right)\)

\(\Leftrightarrow S>\frac{10}{40}+\frac{10}{50}+\frac{10}{60}=\frac{37}{60}>\frac{36}{60}=\frac{3}{5}\)⁽²⁾

Từ ⁽¹⁾ và ⁽²⁾ , ta có :

\(\frac{3}{5}< S< \frac{4}{5}hay\frac{3}{5}< \frac{1}{31}+\frac{1}{32}+...+\frac{1}{60}< \frac{4}{5}\)

Nguyen Ribi Nkok Ngok Khôi Bùi nguyễn ngọc dinh Phùng Tuệ Minh Akai Haruma buithianhtho ?Amanda? Nguyễn Thành Trương Nguyễn Ngô Minh Trí

Bài 1 : x=9

Ta có:

\(\frac{31}{2}.\frac{32}{2}.\frac{33}{2}...\frac{60}{2}=\frac{31.32.33...60}{2^{30}}\)

                                    \(=\frac{\left(31.32.33...60\right).\left(1.2.3...30\right)}{2^{30}.\left(1.2.3...30\right)}\)

                                    \(=\frac{1.2.3...60}{2^{30}\left(1.2.3...30\right)}\)

                                    \(=\frac{\left(1.3.5.7...59\right)\left(2.4.6.8...60\right)}{\left(2.4.6.8...60\right)}\)

                                    \(=1.3.5.7...59\)

Vậy \(\frac{31}{2}.\frac{32}{2}.\frac{33}{2}...\frac{60}{2}=1.3.5.7...59\)