Question

cho ( x₁p - y₁q )²ⁿ + ( x₂p - y₂q )²ⁿ + ( x₃p - y₃q )²ⁿ + ... + ( x_mp - y_mq )²ⁿ \(\le\)0 với m,n thuộc N*

CMR : \(\frac{x_1+x_2+x_3+...+x_m}{y_1+y_2+y_3+...+y_m}=\frac{q}{p}\)

Answer 1

( x₁p - y₁q )²ⁿ \(\ge\)0 ; ( x₂p - y₂q )²ⁿ \(\ge\)0 ; ... ; ( x_mp - y_mq )²ⁿ \(\ge\)0

vậy ( x₁p - y₁q )²ⁿ + ( x₂p - y₂q )²ⁿ  + ... + ( x_mp - y_mq )²ⁿ \(\ge\) 0

mà ( x₁p - y₁q )²ⁿ + ( x₂p - y₂q )²ⁿ  + ... + ( x_mp - y_mq )²ⁿ \(\le\)0

suy ra x₁p - y₁q = x₂p - y₂q = ... = x_mp - y_mq = 0

do đó : \(\frac{x_1}{y_1}=\frac{x_2}{y_2}=...=\frac{x_m}{p_m}=\frac{q}{p}\)hay \(\frac{x_1+x_2+...+x_m}{y_1+y_2+...+y_m}=\frac{q}{p}\)