Question

Cum se face suma lui Gauss cu numere impare ? Explicati ,va rog, si dati un exemplu. Multumesc!

Answer 1

Formula lui Gauss pentru sume de numere impare (suma incepe cu numarul 1)
1 + 3 + 5 + 7 + … + ( 2n – 1 ) = n x n

99 = 2n-1,  de unde 2n = 99 + 1=100 => n = 100 : 2 = 50
deci suma S = 50 x 50 = 2500

Answer 2

$1+3+5+...+99 = \sum\limits_{k=1}^{50}(2k-1) = \sum\limits_{k=1}^{50}(2k) - \sum\limits_{k=1}^{50}(1) = \\ \\ = 2\cdot \sum\limits_{k=1}^{50}(k)-(1+1+\underset{\text{de 50 ori}}{\underbrace{...}}+1) = 2\cdot (1+2+3+...+50)-50 = \\ \\ = 2\cdot \dfrac{50\cdot (50+1)}{2}-50 = 50\cdot (50+1)-50 = 50\cdot 51-50 = \\ \\ = 50\cdot (51-1) = 50\cdot 50 = 50^2 = 2500$