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Ta có : A = 1.2.3 + 2.3.4 + 4.5.6 + ..... + 98.99.100
=> 6A = 1.2.3.4 - 1.2.3.4 + 2.3.4.5 - 2.3.4.5 + ...... + 98.99.100.101
=> 6A = 98.99.100.101
=> A = \(\frac{98.99.100.101}{6}=16331700\)
có 20172 đồng dư 1 mod (3)
=> (20172)50 đồng dư 1 mod (3)
=> (20172)50-1 đồng dư 1-1 = 0 mod (3)
=> dpcm
Tính
a,1.2.3+2.3.4+3.4.5+......+ 98.99.100
b,1 bình +2 bình +3 bình +....+100 bình
Giải:Đặt A=1.2.3+2.3.4+..........+98.99.100
4A=1.2.3.4+2.3.4.5-1.2.3.4+...........+98.99.100.101-97.98.99.100
4A=98.99.100.101=97990200\(\Rightarrow A=24497550\)
b,Đặt B=12+22+................+1002
B=1.(2-1)+2.(3-1)+.............+100.(101-1)
B=1.2+2.3+.......+100.101-1-2-..........-100
Đặt C=1.2+2.3+........+100.101
3C=1.2.3+2.3.4-1.2.3+........+100.101.102-99.100.101
3C=100.101.102=1030200\(\Rightarrow C=343400\)
\(\Rightarrow B=343400-\frac{100.101}{2}=343400-5050=338350\)
c, 4C= (1.2.3+2.3.4+3.4.5+...+8.9.10) .4
==> 4C= [1.2.3.(4-0) + 2.3.4-(5-1) + 8.9.10.(11-7)
==>4C= 1.2.3.4 - 1.2.3.4+ 2.3.4.5-2.3.4.5 + 7.8.9.10- 7.8.9.10 + 8.9.10.11
==> 4C= 8.9.10.11=7920
==> C= 7920 :4=1980
a, Ta có: 3A= 1.2.3+2.3.3+3.4.3+...+99.100.3
3A=1.2.(3-0) + 2.3.(4-1)+ 3.4.(5-2)+ ... + 99.100.( 101-98)
3A=(1.2.3 + 2.3.4+ 3.4.5+ 99.100.101) - (0.1.2 +1.2.3+ 2.3.4 + ... + 98.99.100)
3A= 99.100.101 - 0.1.2
3A= 999900 - 0
3A= 999900
==> A= 999900 : 3
==> A= 333300
a) \(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{98.99.100}\)
\(A=\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{98.99.100}\)
\(A=\left(\frac{1}{1.2}-\frac{1}{2.3}\right)+\left(\frac{1}{2.3}-\frac{1}{3.4}\right)+...+\left(\frac{1}{98.99}-\frac{1}{99.100}\right)\)
\(A=\frac{1}{1.2}-\frac{1}{99.100}\)
\(A=\frac{1}{2}-\frac{1}{9900}\)
\(A=\frac{9898}{19800}.\)
Vậy :
\(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{98.99.100}\)
\(A=\frac{9898}{19800}:2\)
\(A=\frac{4949}{19800}.\)
a) A = \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{98.99.100}\)
A = \(\frac{1}{2}.\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{98.99.100}\right)\)
A = \(\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{98.99}-\frac{1}{99.100}\right)\)
A = \(\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{99.100}\right)\)
A = \(\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{9900}\right)\)
A = \(\frac{1}{2}.\frac{4949}{9900}\)
A = \(\frac{4949}{19800}\)