Question

Rezolvati ecuatia
(X-1)^2+(x-2)^2+....+(x-1000)^2=(x+1)^2+(x+2)^2+....+(x+1000)^2 .
^2 - la puterea a 2 a

Answer 1

suma  1000 termeni  = suma  1000 termeni 
( x - 1) ² - ( x+1 )² = ( x -1 +x +1 ) · ( x -1  -x -1 ) = 2x· ( -2) 
( x -2 )² - ( x +2 )² = ( x -2 +x+2 )· ( x -2  -x  -2 ) =  2x· (  -4) 
trecand totii termenii in partea stanga 
......................................
( x  -1000 )² - ( x +1000 )² = ( x -1000 +x+ 1000 )  ( x -1000  -x -1000) = 2x· ( -2000)
ex . devine 
2x(-2) +2x( -4) + 2x( -6 ) +.......... + 2x( -2000 ) =0 
2x · ( -2-4-6-8 ............ - 2000 ) =0 ⇒  2x=0 ⇒   x =0