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\(\frac{1}{2^2}< \frac{1}{1}-\frac{1}{2}\)
\(\frac{1}{3^2}< \frac{1}{2}-\frac{1}{3}\)
\(.......\)
\(\frac{1}{50^2}< \frac{1}{49}-\frac{1}{50}\)
\(\Rightarrow A< \frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)
\(\Rightarrow A< \frac{1}{1}-\frac{1}{50}=\frac{49}{50}\)
Mà \(\frac{49}{50}< 2\)
\(\Rightarrow A< 2\)
\(A=\frac{1}{1^1}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}\)
\(A=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}\)
Ta thấy \(\frac{1}{2^2}< \frac{1}{1.2};\frac{1}{3^2}< \frac{1}{2.3};\frac{1}{4^2}< \frac{1}{3.4};....;\frac{1}{50^2}< \frac{1}{49.50}\)
Khi đó \(A=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}< 1+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+..+\frac{1}{49.50}=B\)
\(B=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)
\(B=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-...+\frac{1}{49}-\frac{1}{50}< 1\)
Vì \(A< 1+B\)mà \(B< 1\)nên \(B+1< 2\)do đó \(A< 2\)
Vậy \(A< 2\)
1/12+1/22+....+1/502<1/1+1/1x2+1/2x3+....+1/49x50=1-1/50=49/50<2
=>A<2(đpcm)
Bạn xem lời giải ở đường link sau nhé:
Câu hỏi của nguyenducminh - Toán lớp 6 - Học toán với OnlineMath
A=\(\frac{1}{1^2}\)\(+\frac{1}{2^2}\)\(+\frac{1}{3^2}\)\(+...+\frac{1}{50^2}\)
A<1\(+\frac{1}{1.2}\)\(+\frac{1}{2.3}\)\(+...\frac{1}{49.50}\)
=1+1-\(-\frac{1}{2}\)\(+\frac{1}{2}\)\(-\frac{1}{3}\)\(+...+\frac{1}{49}\)\(-\frac{1}{50}\)
=\(1+1-\frac{1}{50}\)
=\(2-\frac{1}{50}\)\(< 2\)
\(\Rightarrow A< 2\)
\(A<1+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}=1+\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+...+\frac{50-49}{49.50}\)
\(A<1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}=2-\frac{1}{50}<2\)
Ta có: \(\frac{1}{2^2}<\frac{1}{1.2};\frac{1}{3^2}<\frac{1}{2.3};....;\frac{1}{50^2}<\frac{1}{49.50}\)
=>\(A<1+\frac{1}{1.2}+\frac{1}{2.3}+.....+\frac{1}{49.50}\)
=>\(A<1+\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.....+\frac{1}{49}-\frac{1}{50}\)
=>\(A<2-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.....+\frac{1}{49}-\frac{1}{50}=2-\frac{1}{50}<2\)
=>A<2 (đpcm)
\(A=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}\)
\(A=1+\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}\right)=1+B\)( B là biểu thức trong ngoặc )
Xét B
\(B=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}\)
\(B<\frac{1}{1.2}+\frac{1}{2.3}+..+\frac{1}{49.50}\)
\(B<\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)
\(B<\frac{1}{1}-\frac{1}{50}\)
\(B<\frac{49}{50}<1\)
Vậy B < 1
\(\Rightarrow A=1+B<1+1=2\)
\(\Rightarrow A<2\)
cho A=\(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}\)
Chứng minh A < 2
Ta có: 1/2^2 < 1/1.2
1/3^2 < 1/2.3
1/4^2 < 3.4
............
1/50^2 < 1/49.50
=> A= 1 + 1/2^2 + 1/3^2 + ....+1/50^2 < 1 + 1/1.2 + 1/2.3 +....+1/49.50 = 1+ 1-1/2+ 1/2 - 1/3 +. ... + 1/49 -1/50 = 1 + 1 - 1/50 = 2-1/50 < 2
Vậy A < 2 (ĐPCM)
Đặt A=1/12+1/22+1/32+1/42+....+1/502<1/1.2+1/2.3+1/3.4+....+1/49.50
A=1-1/50=50/50-1/50=49/50<2
A=49/50<2 hay 98/100<100/100
Vậy A<2
Ta có
A= 1/2 + 1/22 + 1/32 + 1/42 +.......+ 1/502
=1/2.2 + 1/3.3 + 1/4.4 +.......+ 1/50.50
<1/1.2 + 1/2.3 + 1/3.4 +.......+ 1/49.50
= 1 - 1/2 +1/2 - 1/3 + ...........+ 1/49 -1/ 50
= 1 - 1/50 = 49/50 <100/50=2
Vậy A <
\(A=1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}.\)
\(A=1+\frac{1}{2\cdot2}+\frac{1}{3\cdot3}+\frac{1}{4\cdot4}+.......+\frac{1}{50\cdot50}\)
\(< 1+\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+....+\frac{1}{49\cdot50}.\)
\(\Rightarrow1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)
\(1+1-\frac{1}{50}< 2\)
=>A<2
ok xong
A = \(\frac{1}{1^2}\) + \(\frac{1}{2^2}\) + \(\frac{1}{3^2}\)+ \(\frac{1}{4^2}\) + .... + \(\frac{1}{50^2}\)
A = 1 + \(\frac{1}{2.2}\)+ \(\frac{1}{3.3}\)+ \(\frac{1}{4.4}\)+ ...... + \(\frac{1}{50.50}\)< 1 + \(\frac{1}{1.2}\)+ \(\frac{1}{2.3}\)+ \(\frac{1}{3.4}\)+ ...... + \(\frac{1}{49.50}\)
A < 1 + ( 1 - \(\frac{1}{2}\)+ \(\frac{1}{2}\)- \(\frac{1}{3}\)+ \(\frac{1}{3}\)- \(\frac{1}{4}\)+ ...... + \(\frac{1}{49}\)- \(\frac{1}{50}\))
A < 1 + ( 1 - \(\frac{1}{50}\))
A < 1 + 1 - \(\frac{1}{50}\)
A < 2 - \(\frac{1}{50}\)
=> A < 2
\(\frac{1}{2^2}< \frac{1}{1.2}\)
...................\(\frac{1}{50^2}< \frac{1}{49.50}\)
\(\Rightarrow A< \frac{1}{1.2}+\frac{1}{2.3}+....+\frac{1}{49.50}\)
\(\Rightarrow A< 1-\frac{1}{50}< \frac{49}{50}< 1< 2\)
1/2^2<1/1*2;1/3^2<1/2*3;1/4^2<1/3*4;1/50^2<1/49*50
ta có:
=> 1/1^2+1/2*3+1/3*4+...+1/49*50
<=> 1/1-1/2+1/2-1/3+1/3-1/4+...+1/49-1/50
<=> 1-1/50 < 2
=> A < 2