Question
If $${x_1},{x_2},.....,{x_n}$$ are any real numbers and $$n$$ is any positive integer, then
A.
$$n\sum\limits_{i = 1}^n {{x_i}^2 < {{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^2}} $$
B.
$$\sum\limits_{i = 1}^n {{x_i}^2 \geqslant {{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^2}} $$
C.
$$\sum\limits_{i = 1}^n {{x_i}^2 \geqslant n{{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^2}} $$
D.
none of these
Answer :
none of these
Solution :
If any of the inequations hold, it must hold for any real numbers $${x_1},{x_2},.....,{x_n}\,$$ and any $$n \in N.$$
∴ let $${x_1} = 1,{x_2} = 2,{x_3} = 3;n = 3$$ then we can check none of the inequalities (A), (B) or (C) are satisfied.