IIT JEE 1982 Maths - MCQ Question 10

Algebra Level 4

If x 1 , x 2 , x 3 , … , x n x_1, x_2, x_3, \ldots , x_n are any real numbers and n is any positive integer, then which of the option is true?

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n Σ i = 1 n x i 2 < ( Σ i = 1 n x i ) 2 n\Sigma_{i=1}^nx_i^2<(\Sigma_{i=1}^nx_i)^2 None of the others Σ i = 1 n x i 2 ≥ n ( Σ i = 1 n x i ) 2 \Sigma_{i=1}^nx_i^2≥n(\Sigma_{i=1}^nx_i)^2 Σ i = 1 n x i 2 ≥ ( Σ i = 1 n x i ) 2 \Sigma_{i=1}^nx_i^2≥(\Sigma_{i=1}^nx_i)^2

Shubhamkar Ayare by Shubhamkar Ayare , 22, India

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1 solution

Shubhamkar Ayare
Dec 1, 2016

By Generalized Mean Inequality, we directly get n Σ i = 1 n x i 2 ≥ ( Σ i = 1 n x i ) 2 n\Sigma_{i=1}^nx_i^2≥(\Sigma_{i=1}^nx_i)^2 . Hence the answer is 'None of the others'.

The statement can also be made to follow from Cauchy-Schwarz Inequality by setting each b i = 1 b_i=1 .

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