Răspuns:
Explicație pas cu pas:
a = 1 + 3 + 5 + ... + 99
b = 2 + 4 + 6 + ... + 100
_______________________
a) a + b = ?
b) a = pp (pătrat perfect)
a)
a = 1 + 3 + 5 + ... + 99
= (1 + 2 + 3 + ... + 99) - (2 + 4 + 6 + ... + 98)
= [99 × (99 + 1) ÷ 2] - 2 × (1 + 2 + 3 + ... + 49)
= (99 × 100 ÷ 2) - 2 × [49 × (49 + 1) ÷ 2]
= (9900 ÷ 2) - 2 × (49 × 50 ÷ 2)
= 4950 - 2 × (2450 ÷ 2)
= 4950 - 2 × 1225
= 4950 - 2450
= 2500
b = 2 + 4 + 6 + ... + 100
= 2 × (1 + 2 + 3 + ... + 50)
= 2 × [50 × (50 + 1) ÷ 2]
= 2 × (50 × 51 ÷ 2)
= 2 × (2550 ÷ 2)
= 2 × 1275
= 2550
(am aplicat suma lui Gauss-->N × (N + 1) ÷ 2)
a + b = 2500 + 2550
= 5050
b) a = 2500
[tex] \sqrt{2500} = \sqrt{25} \times \sqrt{100 } = 5 \times 10 = 50[/tex]
a = [tex] {50}^{2} [/tex]=> a = pp (pătrat perfect)
