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A=1x2+2x3+3x4+...+49x50
3A= 3(1.2+2.3+3.4+...+49.50)
3A= 1.2.3+2.3.3+3.4.3+...+49.50.3
3A= 1.2.(3-0)+2.3(4-1)+3.4(5-2)+...+49.50.(51-48)
3A= 0.1.2-1.2.3+1.2.3-2.3.4+2.3.4-3.4.5+...+48.49.50-49.50.51
3A= 49.50.51
A= 49.50.51/3=41650
B=1x3+3x5+5x7+...+99x101
B=1/1.3 +1/3.5 +...+1/99.101
2B=2/1.3 + 2/3.5 +...+2/99.101
2B=1-1/3+1/3-1/5+...+1/99-1/101
2B=1-1/101
2B=100/101
B=100/101:2=100/202
\(A=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{99}-\dfrac{1}{100}=\dfrac{1}{2}-\dfrac{1}{100}=\dfrac{49}{100}\)
\(B=\dfrac{1}{2}\left(\dfrac{2}{1\cdot3}+\dfrac{2}{3\cdot5}+...+\dfrac{2}{97\cdot99}\right)\)
\(=\dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{97}-\dfrac{1}{99}\right)\)
\(=\dfrac{1}{2}\cdot\dfrac{98}{99}=\dfrac{49}{99}>\dfrac{49}{100}=A\)
+Câu a:
A = 1/1.2 + 1/2.3 + ...+ 1/5.6 + 1
A = 1/1 - 1/2 + 1/2 - 1/3 + ...+ 1/5 - 1/6 + 1
A = 1/1 - 1/6 + 1
A = 6/6 - 1/6 + 6/6
A = 5/6 + 6/6
A = 11/6
Câu b:
B = 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ...+ 1/98.99.100
B = 1/2. (2/1.2.3 + 2/2.3.4 + ...+ 2/98.99.100)
B = 1/2.(1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 + ...+ 1/98.99 - 1/99.100)
B = 1/2.(1/2 - 1/9900)
B = 1/2.4949/9900
B = 4949/19800
Câu a:
1/1.2 + 1/2.3 + 1/3.4 + 1/4.5 + 1/5.6 + 1
= 1/1 - 1/2 + 1/2 - 1/3 + 1/4 - 1/5 + 1/5 - 1/6 + 1
= 1/1 - 1/6 + 1
= 6/6 - 1/6 + 6/6
= 5/6 + 1
= 11/6
Câu b:
Câu b:
B = 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ...+ 1/98.99.100
B = 1/2. (2/1.2.3 + 2/2.3.4 + ...+ 2/98.99.100)
B = 1/2.(1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 + ...+ 1/98.99 - 1/99.100)
B = 1/2.(1/2 - 1/9900)
B = 1/2.4949/9900
B = 4949/19800
\(M=\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{2}{99.100}\)
\(M=2\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\right)\)
\(M=2\left(1-\frac{1}{100}\right)\)
\(M=2.\frac{99}{100}\)
\(M=\frac{99}{50}\)
\(N=\frac{3}{1.3}+\frac{3}{3.5}+\frac{3}{5.7}+...+\frac{3}{97.99}\)
\(N=\frac{3}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{97}-\frac{1}{99}\right)\)
\(N=\frac{3}{2}\left(1-\frac{1}{99}\right)\)
\(N=\frac{3}{2}.\frac{98}{99}\)
\(N=\frac{49}{33}\)
F = 1- 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/99 - 1/100
= 1 - 1/100
= 99/100
Câu a:
A = 2/1.3 + 2/3.5 + ...+ 2/37.39
A = 1/1 - 1/3 + 1/3 - 1/5 + ...+ 1/37 - 1/39
A = 1/1 - 1/39
A = 39/39 - 1/39
A = 38/39
Câu b:
B = 3/1.4 + 3/4.7 + ..+ 3/97.100
B = 1/1 - 1/4 + 1/4 - 1/7 + ...+ 1/97 - 1/100
B = 1/1- 1/100
B = 100/100 - 1/100
B = 99/100
A = 1x2 + 2x3 + 3x4 + 4x5 + ...+ 99x100
A x 3 = 1x2x3 + 2x3x3 + 3x4x3 + 4x5x3 + ... + 99x100x3
A x 3 = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) + 4x5x(6-3) + ... + 99x100x(101-98)
A x 3 = 1x2x3 + 2x3x4 - 1x2x3 + 3x4x5 - 2x3x4 + 4x5x6 - 3x4x5 + ... + 99x100x101 - 98x99x100.
A x 3 = 99x100x101
A = 99x100x101 : 3
A = 333300
Ta có:
\(A=1.2+2.3+3.4+...+99.100\)
\(\Rightarrow3A=1.2.\left(3-0\right)+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+99.100.\left(101-98\right)\)
\(\Rightarrow3A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100\)
\(\Leftrightarrow3A=99.100.101\Leftrightarrow A=\frac{99.100.101}{3}=333300\)
\(B=1.2.3+2.3.4+4.5.6+...+98.99.100\)
\(\Rightarrow4B=1.2.3.\left(4-0\right)+2.3.4.\left(5-1\right)+4.5.6.\left(7-3\right)+...+98.99.100.\left(101-97\right)\)
\(\Rightarrow4B=1.2.3.4+2.3.4.5-1.2.3.4+4.5.6.7-3.4.5.6+...+98.99.100.101-97.98.99.100\)
\(\Leftrightarrow4B=98.99.100.101\Leftrightarrow B=\frac{98.99.100.101}{4}=24497550\)
Câu a:
A = 1.2 + 2.3 + 3.4 + ...+ 99.100
3A = 1.2.3 + 2.3.3 +..+ 99.100.3
1.2.3 = 1.2.3
2.3.3 = 2.3(4 - 1) = 2.3.4 - 1.2.3
3.4.4 = 3.4(5 - 2) = 3.4.5 - 2.3.4
.............................................................
99.100.3 = 99.100.(101 - 98)=99.100.101-98.99.100
Cộng vế với vế ta có:
3A = 99.100.101
A = 99.100.101 : 3
A = 333300
Câu b:
1.3 + 3.5 + 5.7 + ..+ 97.99
6.A = 1.3.6 + 3.5.6 + 5.7.6 + ...+ 97.99.6
1.3.6 = 1.3.(5 + 1) = 1.3.5 + 1.3.1
3.5.6 = 3.5.(7 - 1) = 3.5.7 - 1.3.5
5.7.6 = 5.7.(9 - 3) = 5.7.9 - 3.5.7
.......................................................
97.99.6 = 97.99.(101 - 95) = 97.99.101-95.97.99
Cộng vế với vế ta có:
6A = 1.3.1 + 97.99.101
A = (1.3.1 + 97.99.101) : 6
A = (3 + 9603.101) : 6
A = (3+ 969903) : 6
A = 969906 : 6
A = 161651
Câu c:
C = 1.2.3 + 2.3.4 + ...+ 98.99.100
4C = 1.2.3.4 +2.3.4.4 + ...+ 98.99.100.4
1.2.3.4 = 1.2.3.4
2.3.4.4 = 2.3.4.(5-1)= 2.3.4.5 - 1.2.3.4
3.4.5.4 = 3.4.5.(6 - 2) = 3.4.5.6 - 2.3.4.5
..........................................................................
98.99.100.4 = 98.99.100.(101-97)=98.99.100.101 - 97.98.99.100
Cộng vế với vế ta có:
4C = 98.99.100.101
C = 98.99.100.101 : 4
C = 24497550
C = 98.99.100.101 :