Question

2.Chứng minh A>1/2 ; còn B<1/2

A= \(\frac{7}{5\times7}+\frac{7}{7\times9}+...+\frac{7}{53\times55}\)

B= \(\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+...+\frac{1}{99}\)

 

Answer 1

A= 7/5*7 + 7/7*9 + ... + 7/53*55

A= 7/2*( 2/5*7 + 2/7*9  +  ... + 2/53*55 )

A= 7/2*( 7-5/5*7 + 9-7/7*9 + ... + 55-53/53*55 )

A= 7/2*( 1/5-1/7 + 1/7-1/9 + ... + 1/53-1/55 )

A= 7/2*( 1/5-1/55 )

A= 7/2*2/11

A= 7/11

A= 7/11 > 1/2

 Nên: A > 1/2

 

B= 1/3 + 1/15 + 1/35 + ... + 1/99

B= 1/1*3 + 1/3*5 + 1/5*7 + ... + 1/9*11

B= 2*( 2/1*3 + 2/3*5 + 1/5*7 + ... + 2/9*11 )

B= 2*( 3-1/1*3 + 5-3/3*5 + 7-5/5*7 + ... + 11-9/9*11 )

B= 2*( 1/1-1/3 + 1/3-1/5 + 1/5-1/7 + ... + 1/9-1/11 )

B= 2*( 1/1-1/11 )

B= 2*10/11

B= 20/11

B= 20/11 < 1/2

Nên: B < 1/2

 

Answer 2

A= 7/5*7 + 7/7*9 + ... + 7/53*55

A= 7/2*( 2/5*7 + 2/7*9  +  ... + 2/53*55 )

A= 7/2*( 7-5/5*7 + 9-7/7*9 + ... + 55-53/53*55 )

A= 7/2*( 1/5-1/7 + 1/7-1/9 + ... + 1/53-1/55 )

A= 7/2*( 1/5-1/55 )

A= 7/2*2/11

A= 7/11

A= 7/11 > 1/2

 Nên: A > 1/2

 

B= 1/3 + 1/15 + 1/35 + ... + 1/99

B= 1/1*3 + 1/3*5 + 1/5*7 + ... + 1/9*11

B= 2*( 2/1*3 + 2/3*5 + 1/5*7 + ... + 2/9*11 )

B= 2*( 3-1/1*3 + 5-3/3*5 + 7-5/5*7 + ... + 11-9/9*11 )

B= 2*( 1/1-1/3 + 1/3-1/5 + 1/5-1/7 + ... + 1/9-1/11 )

B= 2*( 1/1-1/11 )

B= 2*10/11

B= 20/11

B= 20/11 < 1/2

Nên: B < 1/2