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\(\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}.....\frac{899}{30^2}\)
\(=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}.....\frac{29.31}{30.30}=\frac{1.2.3.....29}{2.3.4.....30}.\frac{3.4.5.....31}{2.3.4.....30}\)
\(=\frac{1}{2}.\frac{31}{30}=\frac{31}{60}\)
P \(=\left(1-\frac{1}{2^2}\right).\left(1-\frac{1}{3^2}\right).\left(1-\frac{1}{4^2}\right)...\left(1-\frac{1}{50^2}\right)\)
P\(=\frac{2^2-1}{2^2}.\frac{3^2-1}{3^2}.\frac{4^2-1}{4^2}...\frac{50^2-1}{50^2}\)
P \(=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}...\frac{49.51}{50.50}\)
P\(=\frac{\left(1.2.3...49\right).\left(3.4.5...51\right)}{\left(2.3.4...50\right).\left(2.3.4...50\right)}\)
P\(=\frac{1.51}{50.2}=\frac{51}{100}\)
TL :
Ko biết thì đừng làm
Nhớ làm hết , chi tiết mới đc 1 SP
HT
\(A=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2014}}\)
\(\Rightarrow3A=3+1+\frac{1}{3}+...+\frac{1}{3^{2013}}\)
\(\Rightarrow3A-A\)= \(\left(3+1+...+\frac{1}{3^{2013}}\right)-\left(1+\frac{1}{3}+...+\frac{1}{3^{2014}}\right)\)
\(\Rightarrow2A=3-\frac{1}{3^{2014}}\)
\(\Rightarrow A=\frac{3-\frac{1}{3^{2014}}}{2}\)
\(\Rightarrow A=\frac{3}{2}-\frac{\frac{1}{3^{2014}}}{2}< \frac{3}{2}\)
Vậy \(A< \frac{3}{2}\)
Chúc bạn học tốt !!!
\(A=\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+...+\frac{1}{100}\)
\(A< \frac{1}{10.11}+\frac{1}{11.12}+...+\frac{1}{100.101}\)
\(A< \frac{1}{10}-\frac{1}{11}+\frac{1}{11}-\frac{1}{12}+...+\frac{1}{100}-\frac{1}{101}\)
\(A< \frac{1}{10}-\frac{1}{101}=\frac{101}{1010}-\frac{10}{1010}=\frac{91}{1010}< \frac{505}{1010}\)
\(A< \frac{1}{2}\)