[tex]a_{1} + a_{2} + a_{3} +....+ a_{100} = 100\\a_{1}*2= a_{2}*6 = a_{3}*12=....= a_{100}*10100 = k\\a_{1} = \frac{k}{2} \\a_{2} = \frac{k}{6} = \frac{k}{2*3}\\...\\a_{100} = \frac{k}{100*101} \\[/tex]
[tex]\frac{k}{1*2} + \frac{k}{2*3} + \frac{k}{3*4} + ... + \frac{k}{100*101} = 100[/tex]
[tex]k*(\frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} +\frac{1}{3} - \frac{1}{4} ... + \frac{1}{100} - \frac{1}{101}) = 100[/tex]
Se reduc mare parte din termeni.
Se aduc la acelasi numitor fractiile care raman din paranteza.
[tex]k*\frac{100}{101} = 100\\k = 101[/tex]
[tex]a_{1} = \frac{101}{1*2} \\\\a_{2} = \frac{101}{2*3}\\...\\a_{9} = \frac{101}{9*10} \\[/tex]
Primii noua termeni sunt fractii supraunitare.
100-9 = 91 termeni sunt fractii subunitare
Fractia subunitara este [tex]\frac{a}{b}[/tex] , atunci când a < b
Raspuns 91 termeni
