Question

a=1/3+1/5+5/7+.............2007/2009 b=2/3+2/5+2/7+..............2/2009 aratati ca:a+b/4 apartine N!

Answer 1

a+b=
1/3+3/5+5/7+.............2007/2009+
2/3+2/5+2/7+..............2/2009 = 1/3+2/3+3/5+2/5+....2007/2009+2/2009=
1+1+.....1 de (2009-1)/2 ori adica de 2008/2=1004 oori
Altfel;
daca il scriem pe 1 ca 2*0+1,
3=2*1+1
2007+2*1003+1

deci a+b=1004
 care e dibizibil cu 4, pt ca ultimele doua cifre (04) formeaza un nr. divizibil cu 4
deci 1004/4∈N, cerinta
 aavem numrer de la 0 la 1003 , in total 1004 numere