Question

Calculați:
a) 12 + 15 + 18 + ... + 321
b) 16 + 21 + 26 + ... +521

Answer 1

a) = 3(4+5+6+...107)= suma Gauss cu 107-4+1=104 termeni
    = 3*104(4+107):2 = 3*52*111 = 17 316.
b) progresie aritmetica de ratie r=5 si (521-16):5   +   1 = 505:5  +  1 = 101+1 = 102 termeni
 S= 102(16+521):2 =51*537 = 27 387

Answer 2

a) S = 12 + 15 + 18 + ...+ 321
aflam cati termeni are suma:
(321 - 12) : 3 + 1 = 309 : 3 + 1 = 104

S = 104 × (12 + 321) : 2
S = 104 × 333 : 2
S = 52 × 333
S = 17 316

sau

S = 3×1+9+3×2+9+3×3+9+...+3×104+9

S = 3×(1+2+3+...+104) + 9+9+...+9 de 104 ori

S = 3× 104×105 : 2 + 9×104

S = 3×52×105 + 936

S = 16 380 + 936

S = 17 316


b)
S = 16 + 21 + 26 + ...+ 521
aflam cati termeni are suma:
(521 - 16) : 5 + 1 = 505: 5 + 1 = 102
S = 102 × (16 + 521) : 2
S = 102 × 537 : 2
S = 51 × 537
S = 27 387

sau

S = 5×1+11 + 5×2+11 + 5×3+11 +...+5×102+11

S = 5×(1+2+3+...+102) + (11+11+11+...+11 de 102 ori)

S = 5 × 102 × (102+1) : 2 + 11 × 102

S = 510 × 103 : 2 + 1 122

S = 255 × 103 + 1 122

S = 26 265 + 1 122

S = 27 387