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Ta có:
A = 1 + 3 + 5 + 7 +... + 101
A = \(\frac{102.51}{2}=2601\)
M = 16 - 18 + 20 - 22 + 24 - 26 + .. + 64 - 66 + 68
M = ( 16 - 18 ) + ( 20 - 22 ) + ( 24 - 26 ) + ... + ( 64 - 66 ) + 68
M = (- 2 + - 2 + -2 + ... + - 2 ) + 68
M = 25/2 . ( - 2 ) + 68
M = -25 + 68
M = 43
H = ( 1 + 2 + 3 +...+ 99 ) x ( 13 x 15 - 12 x 15 - 15 )
H = ( 1 + 2 + 3 +...+ 99 ) x { (13 - 12 - 1) x 15 }
H = ( 1 + 2 + 3 +...+ 99 ) x 0
H = 0
G = 1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + 9 + 10 - 11 - 12 + 13 + 14 - ... + 301 + 302
G = ( 1 + 2 ) + ( -3 - 4 ) + ( 5 + 6 ) + ( -7 - 8 ) + ( 9 + 10 ) + ( - 11 - 12 ) + ( 13 + 14 ) -...+ ( 301 + 302 )
G = ( 3 - 7 ) + ( 11 - 15 ) + ( 19 - 23 ) + 27 - ... + 603
G = -4 + - 4 + -4 + 27 - ... + 603
G = 75 x ( -4 ) + 603
G = -300 + 603
G = 303
2.
a) 1 + 2 + 3 + 4 +...+ 99 + 100 + 2 x X = 5052
= > \(\frac{100.101}{2}\)+ 2 x X = 5052
= > 5050 + 2 x X = 5052
= > 2X = 2
= > X = 1
b=1-4+5+6+7+8+9+10+11+12+...+301+302
b=-3+302
b=299
1+2-3-4+5+6-7-8+9+9+10-11-12+12+...229-300+301+302
=(302+1)-(301+2)-(300+3)+(299+4)-...
=303-303-303-303-....
=0-0-0-0-0-....
=0
1+2-3-4+5+6-7-8+9+10-11-12+...-229-300+301+302
= (1+2-3)+(4+5+6-7)+...+(228-229-300+301)+302
= 0 + 0 +...+ 0 + 302
= 302
Ta chia thành: 1 + (2-3-4+5) + (6-7-8+9)+....+(298-299-300+301) + 302 = 1 + 0 + 0 + 0 + .....+ 0 + 302 = 1+302 = 303
A = 1 + ( 2 - 3 - 4 + 5 ) + ( 6 - 7 - 8 + 9 ) + ( 10 - 11 - 12 + 13 ) + ... + ( 298 - 299 - 300 + 301) + 302
A = 1 + 0 + 302
A = 303
a, A=50(100+2):2-49(97+1):2=149
b, B=(1-3)+(2-4)+(5-7)+(6-8)+...+(297-299)+(298-300)+301+302
=-2-2-2-...-2+301+302(150 chữ số 2)
=-2.150+603
=330
Đặt biểu thức trên là A
A=(1-3)+(2-4)+(5-7)+(6-8)+.....+(297-299)+(298-300)+301+302
A=301+302-(3-1)-(4-2)-(7-5)-(8-6)-......-(299-297)-(300-298)
------------------------------------------------------------------
có 150 số hạng
A=301+302-(2+2+2+2+..........+2)
------------------------------
50 số hạng
A=301+302-2x150=303
1+2-3-4+5+6-...+301+302
đặt A=(1+2)+(5+6)+...+(301+302)=3+11+19+...+603
A=[(603-3):8+1]x(603+3)=75x606=45450
đặt B=-3-4-7-8-11-13-...-299-300
B=(-3-4)+(-7-8)+(-11-12)+..+(-299-300)
B=-(7+15+23+...+599)
B=-[(599-7):8+1]x(599+7)=45450
ta có:
1+2-3-4+5+6-7-8+...+301+302=A-B=45450-45450=0
1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + ...+ 301 + 302
= 1 + (2 -3 - 4 + 5) + (6 - 7 - 8 + 9) + ...+(298 - 299 - 300 + 301) + 302
= 1 + ( 2+5 - 3 - 4) + (6 + 9 - 7 - 8) + ...+(298 + 301 - 299 - 300) + 302
= 1 + (7 - 7) + (15 - 15) + ...+(599 - 599) + 302
= 1+ 0 + 0 + ...+ 0 + 302 = 303
3G = 3(5/3+8/3^2+11/3^3+...+302/3^100) = 5 + 8/3 + 11/3^2 + ... + 302/3^99
3G - G = ( 5 + 8/3 + 11/3^2 + ... + 302/3^99 ) - ( 5/3+8/3^2+11/3^3+...+302/3^100 )
2G = 5 + 8/3 + 11/3^2 + ... + 302/3^99 - 5/3 - 8/3^2 - 11/3^3 - ... - 302/3^100
2G = 5 + 1 + 1/3 + 1/3^2 + ... + 1/3^98 - 302/3^100 (1)
Đặt B = 1/3 + 1/3^2 + 1/3^3 + ... + 1/3^98
3B = 1 + 1/3 + 1/3^2 + ... + 1/3^97
3B - B = 1 + 1/3 + 1/3^2 + ... + 1/3^97 - 1/3 - 1/3^2 - 1/3^3 - ... - 1/3^98
2B = 1 - 1/3^98
B = 1/2 - 1/3^98 (2)
Từ (1) và (2) => 2G = 5 + 1 + 1/3 + 1/3^2 + ... + 1/3^98 - 302/3^100 = 6 + 1/2 - 1/3^98
=> G = 3 + 1/4 - 1/3^98
ta có 0 < 1/4 - 1/3^98 < 1/2
=> 3 < 3 + 1/4 - 1/3^98 < 3 + 1/2
suy ra 2 + 5/9 < G < 3 + 1/2