Ta có: \(S=\dfrac{1}{2+\sqrt{2}}+\dfrac{1}{3\sqrt{2}+2\sqrt{3}}+\dfrac{1}{4\sqrt{3}+3\sqrt{4}}+...+\dfrac{1}{100\sqrt{99}+99\sqrt{100}}\)

\(=1-\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{2}}-\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{99}}-\dfrac{1}{10}\)

\(=1-\dfrac{1}{10}=\dfrac{9}{10}\)

S = 1 x 2 + 2 x 3 + ... + 99 x 100

3S = 1 x 2 x 3 + 2 x 3 x (4 - 1) + ..... + 99 x 100 x (101 - 98)

3S = 1 x 2 x 3 + 2 x 3 x 4 - 1 x 2 x 3 + .... + 99 x 100 x 101 - 98 x 99 x 100

3S = 99 x 100 x 101 = 999900

S = 999900 : 3 = 333300

3S=1*2*3+2*3*(4-1)+3*4*(5-2)+.......+99*100*(101-98)

3S=1*2*3+2*3*4-1*2*3+3*4*5-2*3*4+..........+99*100*101-98*99*100

S=99*100*11:3

S=333300

Bạn sửa ở S=99*100*11 thanh 99*100*101

Số số hạng :

(100-1):1+1=100(số hạng)

Tổng bằng:

(100+1)x(100:2)=5050

Bài này đâu phải tổng đâu bạn 

S=1*2+2*3+3*4+....+99*100
=> 3S = 1.2.3 + 2.3.3 + 3.4.3 + .... + 99.100.3
3S = 1.2.( 3 - 0 ) + 2.3.( 4 - 1 ) + 3.4.( 5 - 2 ) +....+ 99.100.(101 - 98 ) 
3S = 1.2.3 - 1.2.0 + 2.3.4 - 2.3.1 + 3.4.5 - 3.4.2 + .... + 99.100.101 - 99.100.98 
3S = 99.100.101 
3S = 999900 
=> S = 999900 : 3 = 333300 
Vậy S = 333300 

\(3S=1.2.3+2.3.3+3.4.3+...+99.100.3\)

\(3S=1.2.3.\left(3-0\right)+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+99.100.\left(101-98\right)\)

\(3S=1.2.3-0.1.2+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100\)

\(3S=99.100.101\)

\(S=\frac{99.100.101}{3}\)

\(S=33.100.101\)

S = 1*2+2*3+3*4+...+99*100

3S=1*2(3-0)+2*3(4-1)+3*4(5-2)+...+99*100(101-98)

3S=1*2*3+2*3*4-1*2*3+3*4*5-2*3*4+...+99*100*101-98*99*100

3S=99*100*101

S=(99*100*101):3

S=333 300