Răspuns:
Explicație pas cu pas:
a)
[tex]\frac{1}{n} -\frac{1}{n+1}=\frac{n+1}{n*(n+1)} -\frac{n}{(n+1)*n}=\frac{(n+1)-n}{n*(n+1)}=\frac{1}{n*(n+1)}[/tex]
b)
[tex]\frac{1}{n} -\frac{1}{n+k}=\frac{n+k}{n*(n+k)} -\frac{n}{(n+k)*n}=\frac{(n+k)-n}{n*(n+k)}=\frac{k}{n*(n+k)}[/tex]
c)
[tex]b=\frac{5}{3*8}+ \frac{7}{8*15}+ \frac{9}{15*24}=\frac{5*5}{3*8*5}+ \frac{7}{8*15}+ \frac{3}{15*8}=\frac{25}{8*15}+ \frac{7}{8*15}+ \frac{3}{8*15}=\\\\
=\frac{25+7+3}{8*15} =\frac{35}{8*15} =\frac{7}{8*3} =\frac{7}{24} [/tex]
=>
[tex]b=\frac{7}{24}> \frac{6}{24}=\frac{1}{4}[/tex]
[tex]b=\frac{7}{24}< \frac{8}{24}=\frac{1}{3}[/tex]
=> b∈[tex](\frac{1}{4} ;\frac{1}{3}) [/tex]
