Ta có: \(M=1\cdot2+3\cdot4+\cdots+57\cdot58\)

\(=1\left(1+1\right)+3\left(3+1\right)+\cdots+57\left(57+1\right)\)

\(=\left(1^2+3^2+\cdots+57^2\right)+\left(1+3+\cdots+57\right)\)

Đặt \(A=1^2+3^2+\cdots+57^2\)

\(=1^2+2^2+3^2+4^2+\cdots+57^2+58^2-\left(2^2+4^2+\cdots+58^2\right)\)

\(=\frac{58\left(58+1\right)\left(2\cdot58+1\right)}{6}-2^2\left(1^2+2^2+\cdots+29^2\right)\)

\(=\frac{58\cdot59\cdot117}{6}-4\cdot\frac{29\left(29+1\right)\left(2\cdot29+1\right)}{6}\)

\(=29\cdot59\cdot39-4\cdot\frac{29\cdot30\cdot59}{6}=29\cdot59\cdot39-4\cdot29\cdot5\cdot59\)

=32509

Đặt B=1+3+...+57

Số số hạng của dãy số là:

(57-1):2+1=56:2+1=28+1=29(số)

Tổng của dãy số là: \(B=\left(57+1\right)\cdot\frac{29}{2}=58\cdot\frac{29}{2}=29\cdot29=841\)

M=A+B

=32509+841

=33350

\(N=1^2+3^2+\cdots+57^2\)

=>N=A

=32509

M-N

=33350-32509

=841

Ai giúp đi, làm ơnnnnnnnnnnnnnnnnnnn

\(B=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)

\(B=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{100}-2.\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{100}\right)\)

\(B=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{100}-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{50}\right)\)

\(B=\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}\)

\(B< \frac{1}{60}+\frac{1}{60}+...+\frac{1}{60}\)

\(B< \frac{50}{60}\Leftrightarrow B< \frac{5}{6}\)

= 22.50.51.52 : 6 – 51.50 = 88400 – 2550 = 85850.

\(A=\left(2-1\right).2+\left(4-1\right).4+\left(6-1\right).6+...+\left(100-1\right).100\\ A=2^2-2+4^2-4+6^2-6+...+100^2-100\\ A=\left(2^2+4^2+...+100^2\right)-\left(2+4+...+100\right)\\ A=2^2\left(1+2^2+3^2+...+50^2\right)-\dfrac{\left(100+2\right).50}{2}\\ A=\dfrac{4.50.51.52}{6}-\dfrac{102.50}{2}=85850\)

D = 1.2+3.4+5.6+...+19.20

3D = 1.2(3-0) + 2.3(4-1) ...... 19.20 (21-18)

3D = 1.2.3-0.2.3.4-1 ......... 19.20.21-18

3D = ( 1.2.3+2.3.4+.....+19.20.21) - ( 0.1.2+ 1.2.3+.... + 18.19.20)

3D = 19.20.21

3D = 7980

  D = 2660

A=1.2+2.3+3.4+...+2017.2018

3A=1.2.3+2.3.3+3.4.3+...+2017.2018.3

3A=1.2.3+2.3.(4−1)+3.4.(5−2)+...+2017.2018.(2019−2016)

3A=1.2.3+2.3.4−1.2.3+3.4.5−2.3.4+...+2017.2018.2019−2016.2017.2018

⇒3A=2017.2018.2019

⇒A=2017.2018.20193

A=2017.2018.20193;B=201833=2018.2018.20183

A=2739315938;B=2739316611

⇒A<B

(98x99x100x101)/4

tick nha mình giải chi tiết cho

Đặt S= 1.2 + 2.3 + 3.4 + ...+ 99.100

=>3S = 1.2.3+2.3.3+3.4.3+...+98.99.3+99.100.3

3S= 1.2.3+2.3(4-1)+3.4(5-2)+...+98.99(100-97)+99.100(101-98)

3S= 1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...-97.98.99+99.100.101-98.99.100

3S = 99.100.101  

3S = 3.33.100.101 

 S=33.100.101

    = 333300

Ta có: A = (2 – 1).2 + (4 – 1).4 + (6 – 1).6 + … + (100 – 1).100

A = 22 – 2 + 42 – 4 + 62 – 6 + … + 1002 – 100

A = (22 + 42 + 62 + … + 1002) – (2 + 4 + 6 + … + 100)

A = 22.(12 + 22 + 32 + … + 502) – (100 + 2).50 : 2

A = 22.50.51.52 : 6 – 51.50 = 88400 – 2550 = 85850.