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a) ta có: 2300 = (23)100 = 8100
3200 = (32)100 = 9100
=> 8^100 < 9^100 => 2^300 < 3^200
b) ta có: 2109 > 2108
phần c bn ghi thiếu đề r
a) ta có: 2300 = (23)100 = 8100
3200 = (32)100 = 9100
=> 8^100 < 9^100 => 2^300 < 3^200
b) ta có: 2109 > 2108
phần c bn ghi thiếu đề r
\(A=3+3^2+...+3^{50}\)
\(\Rightarrow3A=3^2+3^3+...+3^{50}+3^{51}\)
\(\Rightarrow3A-A=3^{51}-3\)
\(\Rightarrow2A=3^{51}-3\)
\(\Rightarrow A=\frac{3^{51}-3}{2}\)
\(B=2-2^2+2^3-2^4+...+2^{2019}-2^{2020}\)
\(2B=2^2-2^3+2^4-2^5+...+2^{2020}-2^{2021}\)
\(B+2B=2-2^{2021}\)
\(3B=2-2^{2021}\)
\(B=\frac{2-2^{2021}}{3}\)
\(C=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2008.2009}\)
\(C=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2008}-\frac{1}{2009}\)
\(C=1-\frac{1}{2009}\)
\(C=\frac{2008}{2009}\)
\(D=\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\frac{1}{9.11}\)
\(D=\frac{1}{2}\left(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+\frac{2}{9.11}\right)\)
\(D=\frac{1}{2}\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}\right)\)
\(D=\frac{1}{2}\left(1-\frac{1}{11}\right)\)
\(D=\frac{1}{2}.\frac{10}{11}=\frac{5}{11}\)
Bài 1:
a) Ta có:
\(3^{200}=\left(3^2\right)^{100}=9^{100}\)
\(2^{300}=\left(2^3\right)^{100}=8^{100}\)
Vì \(9^{100}>8^{100}\Rightarrow3^{200}>2^{300}\)
b) Ta có:
\(71^{50}=\left(71^2\right)^{25}=5041^{25}\)
\(37^{75}=\left(37^3\right)^{25}=50653^{25}\)
Vì \(5041^{25}< 50653^{25}\Rightarrow71^{50}< 37^{75}\)
c) Ta có:
\(\frac{201201}{202202}=\frac{201.1001}{202.1001}=\frac{201}{202}\)
\(\frac{201201201}{202202202}=\frac{201.1001001}{202.1001001}=\frac{201}{202}\)
\(\Rightarrow\frac{201201}{202202}=\frac{201201201}{202202202}\)
Bài 2:
a) \(A=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+....+\frac{1}{50^2}\)
Ta có: \(\frac{1}{1^2}=1;\frac{1}{2^2}< \frac{1}{1.2};\frac{1}{3^2}< \frac{1}{2.3};\frac{1}{4^2}< \frac{1}{3.4};....;\frac{1}{50^2}< \frac{1}{49.50}\)
\(\Rightarrow\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}< 1+\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{49.50}\)
\(\Rightarrow A< 1+1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{49}-\frac{1}{50}\)
\(\Rightarrow A< 1+1-\frac{1}{50}\)
\(\Rightarrow A< 2-\frac{1}{50}< 2\)
b) \(B=2^1+2^2+2^3+...+2^{30}\) (Có 30 số hạng)
\(\Rightarrow B=\left(2^1+2^2+...+2^5+2^6\right)+\left(2^7+2^8+2^9+...+2^{12}\right)+...+\left(2^{25}+2^{26}+...+2^{29}+2^{30}\right)\)
(có \(30:6=5\) nhóm)
\(\Rightarrow B=1\left(2^1+2^2+...+2^6\right)+2^6\left(2^1+2^2+...+2^6\right)+.....+2^{24}\left(2^1+2^2+...+2^6\right)\)
\(\Rightarrow B=1.126+2^6.126+2^{12}.126+...+2^{24}.126\)
\(\Rightarrow B=126.\left(1+2^6+2^{12}+...+2^{24}\right)\)
\(\Rightarrow B=21.6.\left(1+2^6+2^{12}+...+2^{24}\right)⋮21\)
\(\Rightarrow B⋮21\)
\(A=\frac{\left(2018+1\right).2018}{2}=2037171\)
\(B=1.2+2.3+3.4+...+2018.2019\)
\(3B=1.2.3+2.3.3+3.4.3+...+2018.2019.3\)
\(3B=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+2018.2019.\left(2020-2017\right)\)
\(3B=1.2.3+2.3.4-1.2.3+...+2018.2019.2020-2017.2018.2019\)
\(3B=2018.2019.2020\)
\(B=\frac{2018.2019.2020}{3}\)
\(B=2743390280\)
Chúc bạn học tốt ~