Question

8+9+10+.......+20,15+16+17+....+60,21+22+23+....90

Answer 1

8+9+10+.......+20 = (1+2+3+.......+20) - (1+2+3+.....7) =

se scrie ca o diferenta a 2 sume Gauss

Suma Gauss are forma S= [n(n+1)]:2

S1 = (1+2+3+.......+20) =[20 ×(20+1)]:2=210

S2 = (1+2+3+.....7) =(7×8):2=28

S = S1-S2 = 182

2.

15+16+17+....+60 =

(1+2+3+.......+60) - (1+2+3+.....+14) =

se scrie ca o diferenta a 2 sume Gauss

Suma Gauss are forma S= [n(n+1)]:2

S1 = (1+2+3+.......+60) =[60 ×(60+1)]:2=1830

S2 = (1+2+3+.....+14) =(14×15):2 = 105

S = S1-S2 = 1830 - 105 = 1725

3.

21+22+23+....90

(1+2+3+.......+90) - (1+2+3+.....+20) =

se scrie ca o diferenta a 2 sume Gauss

Suma Gauss are forma S= [n(n+1)]:2

S1 = (1+2+3+.......+90) =(90 ×91) :2 = 4095

S2 = (1+2+3+.....+20) =(20×21):2 =210

S = S1 - S2= 4095 - 210 = 3885

Answer 2

$8 + 9 + 10 + ... + 20 = \frac{20 \times 21}{2} - \frac{7 \times 8}{2} = \frac{420}{2} - \frac{56}{? - 2} = 210 - 28 = \\ = 182$
$15 + 16 + 17 + ... + 60 = \\ = \frac{60 \times 60}{2} - \frac{14 \times 15}{2} = \\ = \frac{3660}{2} - \frac{210}{2 } = \\ = 1830 - 105 = \\ = 1725$
$21 + 22 + 23 + ... + 90 = \\ = \frac{90 \times 91}{2} - \frac{20 \times 21}{2} = \\ = \frac{8190}{2} - \frac{420}{2} = \\ = 4095 - 210 = \\ = 3885$