đặt B=1/2.3+1/3.4+...+1/49.50

=1/1.2+1/2.3+1/3.4+...+1/49.50

=1-1/2+1/2-1/3+...+1/49-1/50

=1-1/50<1 (1)

Mà 1<2(2)

A =1/1+1/2.2+1/3.3+...+1/50.50<1-1/2+1/2-1/3+...+1/49-1/50 (3)

từ (1),(2),(3) =>A<2

Ta có : \(A=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...........+\frac{1}{50^2}=1+\frac{1}{2^2}+........+\frac{1}{50^2}\)

=> \(A<1+\frac{1}{1.2}+\frac{1}{2.3}+.............+\frac{1}{49.50}\)

=> \(A<1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.........+\frac{1}{49}-\frac{1}{50}\)

=> \(A<2-\frac{1}{50}\Rightarrow A<2\)

Vậy A nhỏ hơn 2

\(\frac{1}{2^2}< \frac{1}{1}-\frac{1}{2}\)
\(\frac{1}{3^2}< \frac{1}{2}-\frac{1}{3}\)
\(.......\)
\(\frac{1}{50^2}< \frac{1}{49}-\frac{1}{50}\)

\(\Rightarrow A< \frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)
\(\Rightarrow A< \frac{1}{1}-\frac{1}{50}=\frac{49}{50}\)
Mà \(\frac{49}{50}< 2\)
\(\Rightarrow A< 2\)
 

a<2 ai k cho mik, mik se k lại hứa thế lun nói là làm

ta có:1/1^2=1/1

1/2^2=1/2*2<1/1*2=1/1-1/2

1/3^2=1/3*3<1/2*3=1/2-1/3

1/4^2=1/4*4<1/3*4

...

1/50^2=1/50*50<1/49*50=1/49-1/50

=>A=1/1-1/50+1

A=99/50<100/50=2

=>A<2

vậy A<2

\(\frac{1}{2^2}< \frac{1}{1.2}\)

...................\(\frac{1}{50^2}< \frac{1}{49.50}\)

\(\Rightarrow A< \frac{1}{1.2}+\frac{1}{2.3}+....+\frac{1}{49.50}\)

\(\Rightarrow A< 1-\frac{1}{50}< \frac{49}{50}< 1< 2\)

1/2^2<1/1*2;1/3^2<1/2*3;1/4^2<1/3*4;1/50^2<1/49*50

ta có:

   =>    1/1^2+1/2*3+1/3*4+...+1/49*50

  <=>   1/1-1/2+1/2-1/3+1/3-1/4+...+1/49-1/50

  <=>   1-1/50 < 2

    =>   A < 2

A=\(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}\)

  =\(1+\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{50.50}< 1+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)

                                                                                         \(< 1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)

                                                                                          \(< 1+1-\frac{1}{50}=\frac{99}{50}< 2\)

                     => \(A< 2\)

a, \(A=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\)

\(=1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\)

Ta có: \(\frac{1}{2^2}< \frac{1}{1.2};\frac{1}{3^2}< \frac{1}{2.3};...;\frac{1}{50^2}< \frac{1}{49.50}\)

\(\Rightarrow1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}< 1+\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{49.50}\)

\(\Rightarrow1< 1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}< 1+\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{49.50}\)

Mà \(1+\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{49.50}=1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}=1+1-\frac{1}{50}=2-\frac{1}{50}< 2\)

\(\Rightarrow1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}< 2\Rightarrow A< 2\left(đpcm\right)\)

b, B = 2 + 2² + 2³ +...+ 2³⁰

= (2+2²+2³+2⁴+2⁵+2⁶)+...+(2²⁵+2²⁶+2²⁷+2²⁸+2²⁹+2³⁰)

= 2(1+2+2²+2³+2⁴+2⁵)+...+2²⁵(1+2+2²+2³+2⁴+2⁵)

= 2.63+...+2²⁵.63

= 63(2+...+2²⁵)

Vì 63 chia hết cho 21 nên 63(2+...+2²⁵) chia hết cho 21 

Vậy B chia hết cho 21

Cảm ơn bn nhìu nha !!! 

TA CÓ Vế trái <\(\frac{1}{1}+\frac{1}{1.2}+\frac{1}{2.3}+....+\frac{1}{49.50}\)\(=1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.....+\frac{1}{49}-\frac{1}{50}\)

\(=2-\frac{1}{50}< 2\)

do đó VT <2(dpcm)

1+12=13

\(\frac{1}{1^2}=1\)

\(\frac{1}{2^2}< \frac{1}{1\cdot2}\)

\(\frac{1}{3^2}< \frac{1}{2\cdot3}\)

  \(.\)         \(.\)

  \(.\)         \(.\)

  \(.\)         \(.\)

\(\frac{1}{50^2}< \frac{1}{49\cdot50}\)

Cộng vế với vế ta có \(:\)

\(C< 1+\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...........+\frac{1}{49\cdot50}\)

\(C< 1+\frac{49}{50}< \frac{50+49}{50}=\frac{99}{50}< \frac{100}{50}=2\)

Vậy \(C=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...........+\frac{1}{50^2}< 2\)