F \(=1.100+2.\left(100-1\right)+3.\left(100-2\right)+...+100\left(100-99\right)\)

    \(=1.100+2.100-1.2+3.100-2.3+...+100.100-99.100\)

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

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

=> F = 171700

đúng cái nhe

=171700

171700 nhé bạn

F = 1.100 + 2. ( 100 - 1 ) + 3. ( 100 -2 ) + ... + 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 )

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

Vậy F = 171700

F = 1.100 + 2.99 + 3.98 + ... + 98.3 + 99.2 + 100.1 = 1.100 + 2.(100-1) + 3.(100-2) + ... + 99.(100-98) + 100.(100-99) = 100.( 1 + 2 + 3 +... +99 + 100) - ( 1.2 + 2.3 + 3.4 + ... + 98.99 + 99.100) \(=100.\dfrac{\left(1+100\right).100}{2}-A\) = 100. 5050 - A Xét dạng tổng quát của A là B = 1.2 + 2.3 + 3.4 + ...+ n.(n+1) Ta có:
3.B = 1.2.(3 - 0) + 2.3.(4 - 1) + 3.4.(5 - 2) +...+ n.(n+1).((n+2) - (n-1))
= 1.2.3 + 2.3.4 + 3.4.5 + ... + (n-1).n.(n+1) + n.(n+1).(n+2) - 0.1.2 - 1.2.3 - 2.3.4 - 3.4.5 - ... - (n-1).n.(n+1)
=n.(n+1).(n+2) \(\Rightarrow B=\dfrac{n\left(n+1\right).\left(n+2\right)}{3}\) \(\Rightarrow A=\dfrac{99.100.101}{3}=333300\) Vậy F = \(505000-333300=171700\)

D = 1.2 + 2.3 + 3.4 + ... + 99.100

3D = 1.2.3 + 2.3.3 + 3.4.3 + ... + 99.100.3

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

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

3D = 99.100.101

D = 99.100.101 : 3 = ...

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