Question

1)Arătați că :
a)1/2< 1/20+ 1/21+ 1/22+ ...+ 1/39< 1
b)1 / 2la 2 +1/3la 2+1/4la 2+...+1/401la 2 <400/401

2)Determinați valorile lui n apatine N pentru care avem:
a)1/4 <(n+1)la 2 /24 <5/3
b)1/4 <(2n+1)la 2/36 <7/3
c)1/4 <(3n-1) la 2 / 48 <5/3

Answer 1

1)a) $\frac{1}{40} \ \textless \ \frac{1}{20}$
       $\frac{1}{40} \ \textless \ \frac{1}{21}$
       $\frac{1}{40}\ \textless \ \frac{1}{22}$
....................................................
       $\frac{1}{40} \ \textless \ \frac{1}{39}$
                                                                                  +
$\frac{20}{40} \ \textless \ \frac{1}{20}+ \frac{1}{21}+ \frac{1}{22} +...+ \frac{1}{39}$

$\frac{1}{2} \ \textless \ \frac{1}{20}+ \frac{1}{21}+ \frac{1}{22} +...+ \frac{1}{39}$ (1)

$\frac{1}{20}$≤$\frac{1}{20}$

$\frac{1}{21}\ \textless \ \frac{1}{20}$

$\frac{1}{22}\ \textless \ \frac{1}{20}$
.........................................
$\frac{1}{39}\ \textless \ \frac{1}{20}$
                                                                              +
$\frac{1}{20} + \frac{1}{21} +....+ \frac{1}{39} \ \textless \ \frac{20}{20} =1$ (2)
din (1) si (2) ⇒cerinta

b)$\frac{1}{2*2}\ \textless \ \frac{1}{1*2}= \frac{1}{1} - \frac{1}{2}$
 
  $\frac{1}{3*3}\ \textless \ \frac{1}{2*3}= \frac{1}{2} - \frac{1}{3}$

$\frac{1}{4*4}\ \textless \ \frac{1}{3*4}= \frac{1}{3} - \frac{1}{4}$
.............................

$\frac{1}{401*401}\ \textless \ \frac{1}{400*401}= \frac{1}{400} - \frac{1}{401}$
                                                                                                              +
$\frac{1}{ 2^{2} } + \frac{1}{ 3^{2} } + \frac{1}{ 4^{2} } +...+ \frac{1}{ 401^{2} } \ \textless \ \frac{1}{1} - \frac{1}{401} = \frac{401-1}{401} = \frac{400}{401}$

2)a) $\frac{1}{4}\ \textless \ \frac{( n+1)^{2} }{24}\ \textless \ \frac{5}{3}$
Inmultim totul cu 24

$\frac{24}{6}\ \textless \ ( n+1)^{2} \ \textless \ \frac{5*24}{3}$]

$4\ \textless \ ( n+1)^{2} \ \textless \ 40$⇒n=2,3,4,5