Question

sa se arate ca numerele N=1997+2×(1+2+3+...+1996) si P=1+3+5+...+1997 sunt patrate perfecte. Generalizare.

Answer 1

N = 1997 + 2·1996·1997/2 = 1997(1+1996) = 1997²
generalizare: N = (n+1) + 2·(1+2+3+.....+n) = (n+1) +2·n(n+1)/2 = (n+1)(n+1)
P = 1+3+5+......+1997 = 1+(1+2·1)+(1+2·2)+......+(1+2·993)
P = 1·994 + 2(1+2+3+....+993) = 994+ 2·993·994/2 = 994(1+993) = 994²
generalizare:  1+3+ 5 +......+ 2n+1 = 1·(2n+1) + 2·(1+2+......n)/2
P = n+1 + n(n+1) = (n+1)(n+1) = (n+1)²