Question

Rezolvati in ΙΝ ecuatia:
x+3x+5x+....+99x=2x+4x+....+100x-50.

Answer 1

x(1+3+5+...+99)=2x(1+2+3+...50)-50

folosim formula generala pentru Suma lui Gauss:

1+2+3+...+n=$\frac{n(n+1)}{2}$

respectiv pentru suma numerelor impare consecutive:

1+3+5+...+(2k-1)=$k^{2}$

Deci exercitiul devine:

x[1+3+5+...+(2*50-1)]=2x*$\frac{50*51}{2}$ - 50

x*$50^{2}$=x*50*51 - 50

50=x*50(51-50)
50=x*50
x=1