ta có:
4s=1.2.3.(4-0)+2.3.4.(5-1)+3.4.5.(6-2)+.........+k(k+1)(k+2)((k+3)-(k-1))
4s=1.2.3.4-1.2.3.0+2.3.4.5-1.2.3.4+3.4.5.6-2.3.4.5+........+k(k+1)(k+2)(k+3)-(k-1)k(k+1)(k+2)
4s=k(k+1)(k+2)(k+3)
ta biết rằng tích 4 số tự nhiên liên tiếp khi cộng thêm 1 luôn là 1 số chính phương
=>4s+1 là 1 số chính phương

cố gắng lên

ta co:1/1*2*3=(1/1*2-1/2*3):2
1/2*3*4=(1/1*2-1/2*3):2
...
cu nhu the cho den:
1/98*99*100=(1/98*99-1/99*100):2
suy ra : 1/1*2*3+1/2*3*4+1/3*4*5+...+1/98*99*100
=(1/1*2-1/2*3):2+(1/2*3-1/3*4):2+...+(1/98*99-1/99*100):2
=(1/1*2-1/2*3+1/2*3-1/3*4+...+1/98*99-1/99*100):2
=(1/1*2-1/99*100):2
=(1/2-1/9900)
=(4950/9000-1/9000):2
=4949/9000:2
=4949/18000
học tốt

A = 1.3 + 3.5 + ..+ 99.101

6A = 1.3.6 + 3.5.6 + 5.7.6 + ...+99.101.6

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

.........................................................

99.101.6 = 99.101.(103 - 97) = 99.101.103-97.99.101

Cộng vế với vế ta có:

6A = 1.3.1 + 99.101.103

6A = 3 + 9999.103

6A = 3 + 1029897

6A = 1029900

A = 1029900 : 6

A = 171650

Câu b:

B = 1.2.3 + 2.3.4 + 3.4.5 + ...+99.100.101

4B = 1.2.3.4 + 2.3.4.4 + 3.4.5.4 +..+ 99.100.101.4

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

..............................................................................

99.100.101.4 = 99.100.101.(102 - 98) = 99.100.101.102 - 98.99.100.101

Cộng vế với vế ta có:

4B = 99.100.101.102

B = 99.100.101.102 : 4

B = 25497450

=1/2-1/3-1/4+1/3-1/4-1/5+1/5-1/6-1/7+...+1/35-1/36-1/37

giao hoán, kết hợp là ra nha

\(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{6.7.8}\)

\(=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{6.7}-\frac{1}{7.8}\)

\(=\frac{1}{1.2}-\frac{1}{7.8}\)

\(=\frac{1}{2}-\frac{1}{56}\)

\(=\frac{28}{56}-\frac{1}{56}=\frac{27}{56}\)

Dấu . là nhân nha

\(\frac{2}{1.2.3}=\frac{1}{1.2}-\frac{1}{2.3}\)

\(\frac{2}{2.3.4}=\frac{1}{2.3}-\frac{1}{3.4}\)

.......................................

\(\frac{2}{6.7.8}=\frac{1}{6.7}-\frac{1}{7.8}\)

S= \(\frac{1}{1.2}-\frac{1}{7.8}=\frac{27}{56}\)

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