Question

Stabiliti paritatea sumelor: a) S₁= 2+4+6+...+198 b) S₂ = 1+3+5+...+175 c)S₃ = 5+9+13+...+101 d) S₄ = 5+10+15+...+85. repede

Answer 1

a )
S₁ = 2 + 4 + 6 + ... + 198 = 2 ( 1 + 2 + ... + 99 ) = 2·99·100÷2 = 99·100 = 9900
S₁ = 9900

b)
S₂ = 1 + 3 + 5 + ... + 175 = 1 + 2 + 3 + 4 + ... + 175 - 2 - 4 - 6 - .... - 174 = 1 + 2 + 3 + .... + 175 - 2(1 + 2 + 3 + .... + 87) = 175·176÷2 - 2·87·88÷2 = 175·88 - 87·88 = 88 ( 175 - 87 ) = 88·88 = 88²
s₂ = 88²

c)
S₃ = 5 + 9 + 13 + ... + 101
r = 4
n = 25
S₃ = a₁·n + r·n·(n-1)÷2 = 5·25 + 4·25·24÷2 = 125 + 1200 = 1325
S₃ = 1325

d)
S₄ = 5 + 10 + 15 + ... + 85 = 5 ( 1 + 2 + 3 + ... + 17 ) = 5·17·18÷2 = 5·17·9 = 765
S₄ = 765