Question

Sa se demonstreze ca 1+3+5+7+...+31 este patrat perfect

Answer 1

$\rm S=1+3+5+7+...+31\\\\S=2\cdot0+1+2\cdot1+1+2\cdot2+1+...+2\cdot15+1\\\\S=2\cdot(1+2+3...+15)+1+1+...+1\\\\S=2\cdot15\cdot16\div2+16\cdot1\\\\S=15\cdot16+16\\\\S=16\cdot(15+1)\\\\S=16\cdot16\\\\S=16^2\\\\Deci,\ S\ este\ patrat\ perfect$

Answer 2

S = (1+31) + (3+29) + (5+27) +........+(15+17) = 32·8 = 2^8 =( 2^4)²= 16² = p.p.