Question

rezolvati in R , ecuatia : 5+ a+1/2+ a+2/3+ a+3/4+ ....+ a+2010/2011 =2015

Answer 1

[tex] \frac{a+1}{2} +...+ \frac{a+2010}{2011} =2010\\ \frac{a+1}{2}-1 +...+ \frac{a+2010}{2011}-1 =0\\ \frac{a-1}{2} +\frac{a-1}{3}+...+ \frac{a-1}{2011} =0\\ (a-1)(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2011})=0\\ a-1=0\\ a=1 [/tex]
a doua paranteza este o suma de numere pozitive deci diferita de zero

Answer 2

vom adauga 1/2+1/3+1/4+...+1/2011 in fiecare membru al egalitatii

5+a+a+...a+1/2+1/2+2/3+1/3+3/4+1/4+...+2010/2011+1/2011
2015+1/2+1/3+1/4+..+1/2011
atunci
5+2010a+2011 =2015+ (1/2+1/3+...+1/2011)
vom nota Sh2011 suma 1/2+1/3+...1/2011 de la Serie armonica, pecare nu o calculeaza nimeni de mana,  doar pe calculator; este o serie divergenta, adica este crescatoare fara limita;in cazul nostru avem primii 2011 termeni ai seriei armonice
atunci
2010a=2015-2016 +Sh2011
si 2010a=(Sh2011)-1

si a=(Sh2011-1)/2010  ∈Q*⊂R

unde Sh2011=1/2+1/3+1/4+...+1/2011