Question

Calculati sumele :

S1 =1-2+3-4+5...+99-100
S2=-2-4-6...-98-100

Answer 1

$\ Folosim formulele: \\ \\ 1+3+5+...+x = (\frac{x+1}{2})^2 \\ \\ 1+2+3+...+x = \frac{x(x+1)}{2} \\ \\ \\ S_1 = 1-2+3-4+5-...-98+99-100 \\ S_1 = 1+3+5+...+99 -2-4-6-...-100 \\ S_1 = 1+3+5+...+99-2(1+2+3+...+50) \\ \\ \RIghtarrow S_1 = (\frac{99+1}{2})^2 - 2\cdot \frac{50(50+1)}{2} \Rightarrow S_1 = (\frac{100}{2})^2 - 2\cdot \frac{50\cdot 51}{2} \Rightarrow \\ \\ \RIghtarrow S_1 = 50^2 - 50\cdot 51 \Rightarrow S_1 = 50(50-51) \Rightarrow S_1 = 50\cdot( -1) \Rightarrow$
$\Rightarrow S_1 = -50 \\ \\ S_2 = -2-4-6-...-100 \Rightarrow S_2 = -2(1+2+3+...+50)\Rightarrow \\ \\ \Rightarrow S_2 = -2\cdot \frac{50\cdot51}{2} \Rightarrow S_2 = -50 \cdot 51 \Rightarrow S_2 = -2550$

Answer 2

S1 =1-2+3-4+5...+99-100

se observa ca 1-2=-1      3-4=-1   99-100=-1 deci avem o suma de (-1) de 50 de ori din 100:2=50
S1=(-1)*50=-50

S2=-2-4-6...-98-100
S2=-2(1+2+3+50)=(-2)*(50+1)*(50-1+1):2
S2=(-1)*51*50=-51*50=-2550