Question

Dau 100 de puncte x= 1/2·3/4·5/6·....... 999/1000 x=?

Answer 1

exemplu : 1 + 2 + 3 + 4 + ... +8+9 + 10 (reorganizarea seriei termenilor)

= (10+1) + (2+9) + (3+8) + (4+7) + (5+ 6)  

= 11 + 11 + 11 + 11 + 11  

= 11 x 5  

= 55  

Deci, Sn = 1 + 2 + 3 + ... (n-3) + (n-2) + (n-1) + n  

Sn = (n+1) + (n-1 +2) + (n-2 + 3) + etc  

Sn = (n + 1) + (n+1) + (n +1) + ...+ (n+1) (n/2) times  

Sn = (n/2)(n+1)  

Sn = (n^2 + n)/2 asta este formula generala

Exemplu: S(10) = ((10^2) + 10)/2  

= (100 + 10)/2  

= 55  

Deci, S(1000) = ((1000)^2 + 1000)/2  

= (1000000 + 1000)/2  

= 1001000/2  

= 500500 => raspunsul