Question

Aratati ca 3 la puterea 49+ 3 la puterea 50 + 3 la puterea 51 + ...3 la puterea 81 nu e patrat perfect. Obs : grupam termenii, dupa ce le aflam ultima cifra si obs a prima (3 sau 4 ) , dau 0

Answer 1

S=3⁴⁹+3⁵⁰+3⁵¹+...+3⁸¹/·3
3S=3⁵⁰+3⁵¹+3⁵²+...+3⁸²
3S-S=3⁸²-3⁴⁹
2S=3⁸²-3⁴⁹
S=[3⁸²-3⁴⁹]/2
⇒u.c[3⁸²]=u.c[3²]=9;
⇒u.c[3⁴⁹]=u.c[3⁹]=3;
⇒u.c[9-3]/2
⇒u.c[6/2]=3⇒3-nu este cifra de patrat perfect;