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M=1/10 + 1/15 + 1/21 +....+ 1/120
M=2/20 +2/30+2/42+....+2/240
M=2/4.5 + 2/5.6 + 2/6.7 +.....+ 2/15.16
M=2.(1/4.5 +......+ 1/15.16)
M=2.(1/4 -1/5 +1/5 - 1/6 +.....+ 1/15 - 1/16)
M=2.(1/4 - 1/16)
M=2.(4/16 - 1/16)
M=2. 3/16
M=6/16=3/8
Có 1/3 = 8/24 < 9/24 = 3/8 =>1/3<M
Có 1/2 = 4/8>3/8 =>1/2 >M
=> 1/3 < M < 1/2
\(A=\frac{1}{15}+\frac{1}{105}+...+\frac{1}{2145}\)
\(=\frac{1}{1.3.5}+\frac{1}{3.5.7}+...+\frac{1}{11.13.15}\)
\(=\frac{1}{4}\left[\frac{4}{1.3.5}+\frac{4}{3.5.7}+...+\frac{4}{11.13.15}\right]\)
\(=\frac{1}{4}.\left[\frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{3.5}-\frac{1}{5.7}+...+\frac{1}{11.13}-\frac{1}{13.15}\right]\)
\(=\frac{1}{4}.\left[\frac{1}{1.3}-\frac{1}{13.15}\right]=\frac{1}{4}.\left[\frac{1}{3}-\frac{1}{195}\right]=\frac{16}{195}\)
\(\frac{3}{x}+\frac{y}{3}=\frac{5}{6}\Rightarrow\frac{3}{x}=\frac{5-2y}{6}\)
\(\Rightarrow x\left(5-2y\right)=18\)
Từ đó tìm được x,y
#)Giải :
\(\frac{1}{15}+\frac{4}{30}+\frac{9}{45}+\frac{16}{60}+...+\frac{81}{135}=\frac{1}{15}+\frac{2}{15}+\frac{3}{15}+...+\frac{9}{15}=\frac{45}{15}=3\)
Dễ ẹc ak :v rút gọn là ra
=(\(\frac{1}{15}\)+\(\frac{4}{30}\)+\(\frac{16}{60}\)+\(\frac{64}{120}\))+(\(\frac{9}{45}\)+\(\frac{36}{90}\))+(\(\frac{25}{75}\)+\(\frac{81}{135}\))
=(\(\frac{8}{120}\)+\(\frac{16}{120}\)+\(\frac{32}{120}\)+\(\frac{64}{120}\))+(\(\frac{18}{90}\)+\(\frac{36}{90}\))+\(\frac{14}{15}\).
=1+\(\frac{3}{5}\)+\(\frac{14}{15}\).
=\(\frac{8}{5}\)+\(\frac{14}{15}\).
=\(\frac{15}{38}\)