Basic Mathematics & Logarithm
Arithmetic Mean and Power Mean
Grade 11
Question:
<p>If <math>\sum_{i=1}^{n} a_i^2 = M</math>, where <math>a_i > 0</math> and if greatest and least values of <math>\left(\frac{\sum_{i=1}^{n} a_i}{\sum_{i=1}^{n} a_i + M}\right)^2</math> are <math>M_1</math> and <math>M_2</math> respectively, then <math>(M_1 - M_2)</math> is</p>
<p>(a) <math>n M</math></p>
<p>(b) <math>(n-1)M</math></p>
<p>(c) <math>(n+2)M</math></p>
<p>(d) <math>(n+1)M</math></p>
Step-by-Step Solution
Key Concept: Apply the inequality between arithmetic mean of powers and power of arithmetic mean to find extreme values.
<p><strong>Solution:</strong> By the inequality relating AM of 2nd powers to the 2nd power of AM:</p><p><math>\frac{a_1^2 + a_2^2 + a_3^2 + ... + a_n^2}{n} \geq \left(\frac{a_1 + a_2 + a_3 + ... + a_n}{n}\right)^2</math></p><p>This gives us:</p><p><math>\frac{M}{n} \geq \left(\frac{\sum_{i=1}^{n} a_i}{n}\right)^2</math></p><p>Therefore, the maximum and minimum values can be determined using this constraint.</p><p>∴ <math>(M_1 - M_2) = (n-1)M</math></p>
Correct Answer: b