A=333300

B=25497450

dễ mà đọc kĩ đi

a)

Ta có : ( 1 + 2 + 3 + ... + 99)

Số số hạng là:       ( 99 - 1 )  : 1 + 1 = 100

Tổng là:                 ( 99 + 1 ) x 100 : 2 = 5000

=> 5000 x ( 13  - 12 - 1 ) x 15

=> 5000 x 10 x 15

=> 50000 x 15

=> 750000

Ko muốn vt nx :))

a:Sửa đề: 1x3+2x4+...+99x101

\(=1\times\left(1+2\right)+2\times\left(2+2\right)+\cdots+99\times\left(99+2\right)\)

\(=\left(1\times1+2\times2+\cdots+99\times99\right)+2\times\left(1+2+\cdots+99\right)\)

\(=\frac{99\times\left(99+1\right)\times\left(2\times99+1\right)}{6}+2\times\frac{99\times100}{2}\)

\(=\frac{99\times100\times199}{6}+99\times100=33\times50\times199+99\times100\)

\(=33\times50\times\left(199+3\times2\right)=33\times50\times205=338250\)

b: \(\frac89\times\frac{15}{16}\times\ldots\times\frac{2499}{2500}\)

\(=\left(1-\frac19\right)\times\left(1-\frac{1}{16}\right)\times\ldots\times\left(1-\frac{1}{2500}\right)\)

\(=\left(1-\frac13\right)\times\left(1-\frac14\right)\times\ldots\times\left(1-\frac{1}{50}\right)\times\left(1+\frac13\right)\times\left(1+\frac14\right)\times\ldots\times\left(1+\frac{1}{50}\right)\)

\(=\frac23\times\frac34\times\ldots\times\frac{49}{50}\times\frac43\times\frac54\times\ldots\times\frac{51}{50}=\frac{2}{50}\times\frac{51}{3}=\frac{17}{25}\)

A=9954

Tính tổng: 1x2 + 2x3 + 3x4 + 4x5 +.............+ 99x100

Gọi biểu thức trên là A, ta có :

A = 1x2 + 2x3 + 3x4 + 4x5 + ...+ 99x100

3A= 1x2x3 + 2x3x3 + 3x4x3 + 4x5x3 + ... + 99x100x3

3A = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) + 4x5x(6-3) + ... + 99x100x(101-98)

3A = 1x2x3 + 2x3x4 - 1x2x3 + 3x4x5 - 2x3x4 + 4x5x6 - 3x4x5 + ... + 99x100x101 - 98x99x100.

3A = 99x100x101

A = 99x100x101 : 3

A = 333300

Ta thấy:

1 x 4 = 1 x 2 + 1 x 2

2 x 5 = 2 x 3 + 2 x 2

3 x 6 = 3 x 4 + 3 x 2 

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

Suy ra:

D = (1 x 2 + 2 x 3 +  3 x 4 + .... + 97 x 98) + (1 x 2 + 2 x 2 + 3 x 2 + .... + 97 x 2)

D = (1x2+2x3+3x4+...+97x98) + (1+2+3+...+99)x2

D = (1x2+2x3+3x4+...+97x98) + 100 x 99 : 2

D  - 100 x 99 : 2 = 1x2+2x3+3x4+...+97x98

D - 4950 = 1x2+2x3+3x4+...+97x98

(D - 4950) x 3 = 1x2x(3-0)+2x3x(4-1)+3x4x(5-2)+......+97x98x(99-96)

(D-4950)x3 = 1 x 2 x 3 + 2 x 3 x 4 - 1 x 2 x 3 + 3 x 4 x 5 - 2 x 3 x 4 + .... + 97 x 98 x 99 - 96 x 97 x 98

(D-4950)x3 = 97 x 98 x 99

Và từ đây ta có thể tìm hướng để ra kết quả

\(A=20\times21+21\times22+...+99\times100\)

\(3\times A=20\times21\times\left(22-19\right)+21\times22\times\left(23-20\right)+...+99\times100\times\left(101-98\right)\)

\(=20\times21\times22-19\times20\times21+...+99\times100\times101-98\times99\times100\)

\(=99\times100\times101-19\times20\times21\)

Suy ra \(A=\frac{99\times100\times101-19\times20\times21}{3}=360640\)

\(B=3\times4\times5+4\times5\times6+...+98\times99\times100\)

\(4\times B=3\times4\times5\times\left(6-2\right)+4\times5\times6\times\left(7-3\right)+...+98\times99\times100\times\left(101-97\right)\)

\(=3\times4\times5\times6-2\times3\times4\times5+...+98\times99\times100\times101-97\times98\times99\times100\)

\(=98\times99\times100\times101-2\times3\times4\times5\)

Suy ra \(B=\frac{98\times99\times100\times101-2\times3\times4\times5}{4}=24497520\)

A=1.2+2.3+3.4+...+99.100

3A=1.2.3+2.3.(4-1)+3.4.(5-2)+...+99.100(101-98)

3A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100

3A=99.100.101

A=333300

333300