Ta có:

\(C= 4+44+444+......+4444444444\)

\(C= 4.(10.1+9.10+8.100+7.1000+...+1.1000000000\)

\(C= 4.(100+90+800+7000+60000+500000+4000000+30000000+200000000+1000000000)\)

\(C=4.12345678900\)

\(C=4938271600\)

Tương tự.

cái này bạn mở sách bồi dưỡng toán ra trang gần cuối là thấy ngay ấy mà

Câu 1: Làm tri tiết để hiểu nha:

Ta có  : S = 3 + 5 + 7 + … + 999

Tổng dãy số tự nhiên lẻ cách đều 2 đơn vị : S = 499*(3 +999) : 2 = 249999

Câu 2:

Ta có: :B=1+11+2+...+99=1+2+..+9+11+22+..+99=9*(1+9):2 + 9*(11+99) :2= 540

Câu 3: 

C= 3+7+11+...+99 = [(99-3)/(7-4) + 1] * (3+99) : 2 =1734

Câu 4:

Ta có D= 1-2+3-4+5-6+...+99-100+101 

= (1+3+5+...+101) - (2+4+6+...+100) 

từ 1 đến 101 có : (101-1):2+1=51 

1+..+101 = (1+101)x 51:2= 2601 

từ  2 đến 100 có : (100-2);2+1=50 

2+...+100 = (100 +2) x 50:2=2550 

=> A= 2601-2550=51

Mình đã gặp bài này rồi .mình sẽ nói cho ban,nhưng bạn phải .Đáp án là:51

a)\(1-2+3-4+5-6+7-8+8-9+9-10\)

=\(\left(1-2\right)+\left(3-4\right)+\left(5-6\right)+\left(7-8\right)+\left(8-9\right)+\left(9-10\right)\)

\(=\left(-1\right)+\left(-1\right)+\left(-1\right)+\left(-1\right)+\left(-1\right)+\left(-1\right)\)

\(=\left(-1\right).6\)

\(=-6\)

b)\(1-2+3-4+...+99-100\)

\(=\left(1-2\right)+\left(3-4\right)+...+\left(99-100\right)\)}\(\left[\left(100-1\right):1+1\right]:2=50\)(cặp)

\(=\left(-1\right)+\left(-1\right)+\left(-1\right)+...+\left(-1\right)\)} 50 số (-1)

\(=\left(-1\right).50\)

\(=-50\)

c)\(1-3+5-7+9-11+13-15\)

\(=\left(1-3\right)+\left(5-7\right)+\left(9-11\right)+\left(13-15\right)\)

\(=\left(-2\right)+\left(-2\right)+\left(-2\right)+\left(-2\right)\)

\(=\left(-2\right).4\)

\(=-8\)

d)\(1-3+5-7+...-99+101\) (Đối với bài này, có vẻ đề sai, mình đã sửa lại rồi

\(=\left(1-3\right)+\left(5-7\right)+...+\left(97-99\right)+101\) } \(\left[\left(99-1\right):2+1\right]:2=25\)(cặp)

\(=\left(-2\right)+\left(-2\right)+\left(-2\right)+...+\left(-2\right)\) } 25 số (-2)

\(=\left(-2\right).25\)

\(=-50\)

e)\(-1-2-3-4-...-99-100\)

\(=\left(-1\right)+\left(-2\right)+\left(-3\right)+...+\left(-99\right)+\left(-100\right)\)

\(=\left[\left(-1\right)+\left(-100\right)\right]+\left[\left(-2\right)+\left(-99\right)\right]+...+\left[\left(-51\right)+\left(-50\right)\right]\) } \(\left[\left(100-1\right):1+1\right]:2=50\)(cặp) (phần này của đề bài, không thay được như (-100) hoặc (-1))

\(=\left(-100\right)+\left(-100\right)+\left(-100\right)+...+\left(-100\right)\)} 50 số (-100)

\(=\left(-100\right).50\)

\(=-5000\)

a, -5

b, -50

c, -8

d, -50

e, -5050

cách làm nữa nha

\(\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+....+\frac{2}{99\cdot101}\)

\(\frac{2}{1\cdot3}=\frac{3-1}{1\cdot3}=\frac{3}{1\cdot3}-\frac{1}{1\cdot3}=\frac{1}{1}-\frac{1}{3}=1-\frac{1}{3}\)

\(\frac{2}{3\cdot5}=\frac{5-3}{3\cdot5}=\frac{5}{3\cdot5}-\frac{3}{3\cdot5}=\frac{1}{3}-\frac{1}{5}\)

....

\(\frac{2}{99\cdot101}=\frac{101-99}{99\cdot101}=\frac{101}{99\cdot101}-\frac{99}{99\cdot101}=\frac{1}{99}-\frac{1}{101}\)

\(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{101}=1-\frac{1}{101}=\frac{100}{101}\)

\(\frac{5}{1\cdot3}+\frac{5}{3\cdot5}+\frac{5}{5\cdot7}+...+\frac{5}{99\cdot101}\)

=\(\frac{5}{2}\cdot\frac{2}{1\cdot3}+\frac{5}{2}\cdot\frac{2}{3\cdot5}+\frac{5}{2}\cdot\frac{2}{5\cdot7}+...+\frac{5}{2}\cdot\frac{2}{99\cdot101}\)

=\(\frac{5}{2}\cdot\left[\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\frac{2}{5\cdot7}+...+\frac{2}{99\cdot101}\right]\)

=\(\frac{5}{2}\cdot\left[1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{99}-\frac{1}{101}\right]\)

=\(\frac{5}{2}\cdot\left(1-\frac{1}{101}\right)\)

=\(\frac{5}{2}\cdot\frac{100}{101}\)

\(=\frac{250}{101}\)

= 3 - 1 / 1 x 3 + 5 - 3 / 3 x 5 + ... + 101 - 99 / 99 x 101

= 1 - 1 / 3 + 1 / 3 - 1 / 5 + 1 / 5 - ... - 1 / 99 + 1 / 99 - 1 / 101

gạch gạch gạch gạch ... gạch gạch

= 1 - 1 / 101

= 100 / 101

A=3.(1/1.2+1/2.3+1/3.4+.....+1/399.400)

A=3.(1/1-1/2+1/2-1/3+......+1/399-1/400)

A=3.(1-1/400)

A=3.399/400

A=1197/400

A=3.(1/1.2+1/2.3+1/3.4+.....+1/399.400)

A=3.(1/1-1/2+1/2-1/3+......+1/399-1/400)

A=3.(1-1/400)

A=3.399/400

A=1197/400

Bài 1: Tính nhanh:

A = 3/1*2 + 3/2*3 + 3/3*4 + ... + 3/399*400

=>3A = 1/1*2 + 1/2*3 + 1/3*4 + ... + 1/399*400

    3A = 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/399 - 1/400

    3A = 1 - 1/400

      3A = 400/400 - 1/400

      3A = 399/400

        A = 399/400 : 3

        A = 399/400 . 1/3

        A = 133/400.

Có gì ko hiểu bn ib mk nha.^^

\(A=\frac{3}{1.2}+\frac{3}{2.3}+\frac{3}{3.4}+...+\frac{3}{399.400}\)

\(A=3.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{399.400}\right)\)

\(A=3.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{399}-\frac{1}{400}\right)\)

\(A=3.\left(1-\frac{1}{400}\right)\)

\(A=3.\frac{399}{400}\)

\(A=\frac{1197}{400}\)

\(B=\frac{5}{1.2}+\frac{5}{2.3}+\frac{5}{3.4}+...+\frac{5}{399.400}\)

\(B=5.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{399.400}\right)\)

\(B=5.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{399}-\frac{1}{400}\right)\)

\(B=5.\left(1-\frac{1}{400}\right)\)

\(B=5.\frac{399}{400}\)

\(B=\frac{399}{80}\)

\(C=\frac{2}{5.7}+\frac{2}{7.9}+\frac{2}{9.11}+...+\frac{2}{149.151}\)

\(C=\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}+...+\frac{1}{149}-\frac{1}{151}\)

\(C=\frac{1}{5}-\frac{1}{151}\)

\(C=\frac{146}{755}\)

\(D=\frac{3}{5.7}+\frac{3}{7.9}+\frac{3}{9.11}+...+\frac{3}{149.151}\)

\(D=\frac{3}{2}.\left(\frac{2}{5.7}+\frac{2}{7.9}+\frac{2}{9.11}+...+\frac{2}{149.151}\right)\)

\(D=\frac{3}{2}.\left(\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}+...+\frac{1}{149}-\frac{1}{151}\right)\)

\(D=\frac{3}{2}.\left(\frac{1}{5}-\frac{1}{151}\right)\)

\(D=\frac{3}{2}.\frac{146}{755}\)

\(D=\frac{219}{755}\)

\(E=\frac{11}{1.3}+\frac{11}{3.5}+\frac{11}{5.7}+...+\frac{11}{99.101}\)

\(E=\frac{11}{2}.\left(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+...+\frac{2}{99.101}\right)\)

\(E=\frac{11}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{99}-\frac{1}{101}\right)\)

\(E=\frac{11}{2}.\left(1-\frac{1}{101}\right)\)

\(E=\frac{11}{2}.\frac{100}{101}\)

\(E=\frac{550}{101}\)

_Chúc bạn học tốt_

\(\frac{2}{1.2}+\frac{2}{3.5}+\frac{2}{5.7}+......+\frac{2}{99.101}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+......+\frac{1}{99}-\frac{1}{100}\)

\(=1-\frac{1}{100}\)

\(=\frac{99}{100}\)

Bạn giúp mk nốt b được ko?