Question

If $a_1,a_2,\....,a_n$ $(n>3)$ are all unequal positive real numbers and \[ E = \frac{(1+a_1+a_1^2)(1+a_2+a_2^2)\dots(1+a_n+a_n^2)}{a_1 a_2 \dots a_n}, \] then which of the following best describes $E$?

• A. $E \le 2^n$
• B. $E \ge 3^n$
• C. $E>3^n$
• D. $E>2^n$

Hint:

Apply AM–GM on each factor, then multiply inequalities. Strict inequality occurs when all terms are not equal.

Answer 1

The Correct Option is C

Step 1: Apply AM ≥ GM for each term

By the AM–GM inequality, for any positive \( a \): \[ 1 + a + a^2 \ \ge \ 3\sqrt[3]{a^2} \]

Step 2: Divide through by \( a \)

\[ \frac{1 + a + a^2}{a} \ \ge \ 3\sqrt[3]{a} \]

Step 3: Apply to all \( n \) terms

For: \[ E = \prod_{i=1}^n \frac{1 + a_i + a_i^2}{a_i} \] we have: \[ E \ \ge \ \prod_{i=1}^n 3\sqrt[3]{a_i} \]

Step 4: Combine cube roots

\[ E \ \ge \ 3^n \cdot \sqrt[3]{\prod_{i=1}^n a_i} \] Since all \( a_i > 0 \), the cubic root is positive.

Step 5: Strict inequality

If all \( a_i \) are not equal, then the AM–GM inequality is strict, so: \[ \boxed{E > 3^n} \]