1)a) [tex] \frac{1}{40} \ \textless \ \frac{1}{20} [/tex]
[tex] \frac{1}{40} \ \textless \ \frac{1}{21} [/tex]
[tex] \frac{1}{40}\ \textless \ \frac{1}{22} [/tex]
....................................................
[tex] \frac{1}{40} \ \textless \ \frac{1}{39} [/tex]
+
[tex] \frac{20}{40} \ \textless \ \frac{1}{20}+ \frac{1}{21}+ \frac{1}{22} +...+ \frac{1}{39} [/tex]
[tex] \frac{1}{2} \ \textless \ \frac{1}{20}+ \frac{1}{21}+ \frac{1}{22} +...+ \frac{1}{39} [/tex] (1)
[tex] \frac{1}{20} [/tex]≤[tex] \frac{1}{20} [/tex]
[tex] \frac{1}{21}\ \textless \ \frac{1}{20} [/tex]
[tex] \frac{1}{22}\ \textless \ \frac{1}{20} [/tex]
.........................................
[tex] \frac{1}{39}\ \textless \ \frac{1}{20} [/tex]
+
[tex] \frac{1}{20} + \frac{1}{21} +....+ \frac{1}{39} \ \textless \ \frac{20}{20} =1[/tex] (2)
din (1) si (2) ⇒cerinta
b)[tex] \frac{1}{2*2}\ \textless \ \frac{1}{1*2}= \frac{1}{1} - \frac{1}{2} [/tex]
[tex] \frac{1}{3*3}\ \textless \ \frac{1}{2*3}= \frac{1}{2} - \frac{1}{3} [/tex]
[tex] \frac{1}{4*4}\ \textless \ \frac{1}{3*4}= \frac{1}{3} - \frac{1}{4} [/tex]
.............................
[tex] \frac{1}{401*401}\ \textless \ \frac{1}{400*401}= \frac{1}{400} - \frac{1}{401} [/tex]
+
[tex] \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} [/tex]
2)a) [tex] \frac{1}{4}\ \textless \ \frac{( n+1)^{2} }{24}\ \textless \ \frac{5}{3} [/tex]
Inmultim totul cu 24
[tex] \frac{24}{6}\ \textless \ ( n+1)^{2} \ \textless \ \frac{5*24}{3} [/tex]]
[tex]4\ \textless \ ( n+1)^{2} \ \textless \ 40[/tex]⇒n=2,3,4,5
