Question

Se considera numarul a=1/5+7/10+8/15+...+50/225-(1/2+1/3+1/4+...+1/45). Aratati ca a este nr natural, patrat perfect. Determinati n→N astfel incat a/2n+1→Z

Answer 1

amplificam toti termenii din a doua paranteza cu 5 si se obtine:

5*(1/2+1/3+1/4+...+1/45)=(5/10+5/15+...+5/225)
=>a=1/5+7/10-5/10+8/15-5/15+...+50/225-5/225=>
=>a=1/5+2/10+3/15+...+45/225=>
=>a=1/5 +1/5 +1/5+....+1/5 =>a=1/5 *45 =9 =3^2 = p.p.

9/2n+1 <=>2n+1=9*k, k e N* <=>n=(9*k-1)/2 <=>n=(9*(2p+1)-1)/2,p eN<=>n=(18p+9-1)/2 <=> n=9p+4, oricare p e N