Question

fie a=1+3+5+7+9+....+23+25. Aratati ca a este patrat perfect

Answer 1

1 + 3 + .. + 2n - 1 = n²

2n - 1 = 25 => 2n = 26 => n = 13

1 + 3 + .. + 25 = 13² = patrat perfect

Answer 2

a = 1 + 3 + 5 + 7 + .... + 23 + 25

→ stabilesc cati termeni are suma cu ratia = 2 ( 3 - 1 = 2;   5 - 3 = 2 .....)

( 25 - 1 ) : 2 + 1 = 24 : 2 + 1 = 12 + 1 = 13 termeni are suma

→ aplic formula sumei lui Gauss

a = 13 × ( 1 + 25 ) : 2

a = 13 × 26 : 2

a = 13 × 13

a = 13² → patrat perfect