Question

Demonstrati ca media aritmetica a primelor k numere naturale pare consecutive si cea a primelor k numere naturale impare consecutive sunt numere naturale consecutive, unde k∈N si k≥2.

ms :)

Answer 1

M pare = (2+4+6+ ......+2k)/k = 2(1+2+3 +.....+k)/k =[ 2·k(k+1)/2]/k = k(k+1)/k = (k+1) ∈ N
Mimp. = (1+3+5+.......+ 2k-1)/k = [(2·0+1)+(2·1+1) +(2·2+1)+........2(k-1)+1]/k =
= [1·k + 2(1+2+3+.......+k-1)]/k =[ k + 2·(k-1)·k/2]/k =( k +k² -k)/k = k ∈N