\(\frac{1+\left(1+2\right)+\left(1+2+3\right)+...+\left(1+2+3+...+100\right)}{100.1+99.2+...+1.100}\)

\(\frac{1+1+2+1+2+3+...+1+2+...+100}{100.1+99.2+...+1.100}\)

\(=\frac{1.100+2.99+3.98+...+100.1}{100.1+99.2+...+1.100}\)

\(=1\)

dễ quớ đi!

Ai ngu mới hỏi câu dễ thế

\(\frac{1+\left[1+2\right]+\left[1+2+3\right]+...+\left[1+2+3+...+100\right]}{100.1+99.2+98.3+...+2.99+1.100}=\frac{1.2:2+2.3:2+3.4:2+...+100.101:2}{100.1+99.2+98.3+...+2.99+1.100}\)

\(=\frac{\frac{1}{2}\left[1.2+2.3+3.4+...+100.101\right]}{100.1+99.2+98.3+...+2.99+1.100}=\frac{\frac{1}{2}\cdot\frac{1}{3}\left[1.2.3-0.1.2+2.3.4-1.2.3+...+100.101.102-99.100.101\right]}{1.100+2.100-1.2+3.100-2.3+...+100.100-99.100}\)

\(=\frac{\frac{1}{6}\cdot100.101.102}{100\left[1+2+3+...+100\right]-\left[1.2+2.3+...+99.100\right]}=\frac{171700}{100\cdot\frac{100.101}{2}-\frac{99.100\cdot101}{3}}\)

\(=\frac{171700}{505000-333300}=\frac{171700}{171700}=1\)

AI THẤY ĐÚNG NHỚ ỦNG HỘ NHÉ

Tử số  \(=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{100}\)

\(=\left(1+\frac{1}{100}\right)+\left(\frac{1}{2}+\frac{1}{99}\right)+...+\left(\frac{1}{50}+\frac{1}{51}\right)\)

\(=\frac{101}{1.100}+\frac{101}{2.99}+...+\frac{101}{50.51}\)

\(=101.\left(\frac{1}{1.100}+\frac{1}{2.99}+...+\frac{1}{50.51}\right)\)

Mẫu số \(=\frac{1}{1.100}+\frac{1}{2.99}+...+\frac{1}{99.2}+\frac{1}{100.1}\)

\(=2.\left(\frac{1}{1.100}+\frac{1}{2.99}+...+\frac{1}{50.51}\right)\)

=> phân số đề bài cho \(=\frac{101}{2}\)

= 99 nha ban

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

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

3N = 99.100.101

3N=33.100.101=333300

b)

tổng này có  99-10+1=90 (số hạng):

10,11 + 11,12 + 12,13 +............+ 98,99 + 99,100 =

10,100 + 11,11 + 12,12 + .......... + 98,98 + 99,99 =

(10,10 + 99,99) x 90 : 2 = 4954,05

c)

R=1.(2-1)+2.(3-1)+.....+100.(101-1)

=1.2-1.1+2.3-1.2+......+100.101-1.100

=(1.2+2.3+.....+99.100+100.101)-(1+2+3+...+100)

=[1.2.3+2.3.(4-1)+........100.101.(102-99)]:3+[(100+1).100:2]

(tổng trên chia cho 3 nên cuối cùng chia 3)

=(1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+.....100.101.102-99.100.101):3+5050

=(100.101.102) :3 +5050

=348450

d)=1.100+2.(100-1)+.....+100.(100-99)

=1.100+2.100-1.2+3.100-2.3+........+100.100-99.100

=100.(1+2+3+.......+100)-(1.2+2.3+3.4+....+99.100)

=100.\(\frac{101.100}{2}-\frac{99.100.101}{3}\) =505000-333300=171700

p/s mỏi tay, bấm mình nhé

a)N=1.2+2.3+3.4+4.5+......+99.100

3N=1.2.3+2.3.3+3.3.4+3.4.5+......+3.99.100

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

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

3N=99.100.101=999900

N=333300

b)P=10.11+11.12+12.13+....+29.30

3P=10.11.3+11.12.3+....+29.30.3

3P=10.11.12-9.10.11+11.12.13-10.11.12+.........+29.30.31-28.29.30

3P=29.30.31-9.10.11=25980

P=8660

R=1²+2²+3²+....+100²

R=1.(2-1)+2.(3-1)+.........+100.(101-1)

R=1.2-1+2.3-2+3.4-3+.....+100.101-100

R=(1.2+2.3+3.4+..........+100.101)-(1+2+3+.........+99+100)

\(R=\left(N+100.101\right)-\frac{100.101}{2}\)

R=343400-5050=338350

S=100.1+2.(100-1)+3.(100-2)+......+99.(100-98)+100(100-99)

S=100.1+100.2-1.2+100.3-2.3+.......+99.100-98.99+100.100-99.100

S=100(1+2+...+100)-(1.2+2.3+....+99.100)

S=100.(100.101:2)-N

S=505000-333300=171700

=1.100 + 2.(100-1) + 3.(100 - 2 ) +..... + 99 .(100 - 98) + 100 .(100 - 99 )

=1.100 + 2. 100 - 1.2 + 3.100 - 3.2 +...+ 99 . 100 - 99 . 98 + 100 .100 - 100.99

= 100 (1 + 2 + 3 +... +100) - (1.2 +3 .2 +...+ 99.98 + 100.99 

=100.100.101 /2 - 99.100.101 / 3 

= 505000 - 333300= 171700 

                                                                Đ/S : 171700