The inequality you're referring to is:

\[
(1 + a_1)(1 + a_2)\dots(1 + a_n) \geq 2^n
\]

This inequality appears to be a form of **AM-GM inequality** (Arithmetic Mean - Geometric Mean inequality). Let's analyze it under the assumption that \(a_1, a_2, \dots, a_n \) are non-negative real numbers.

### Applying the AM-GM Inequality:
The AM-GM inequality states that for non-negative numbers \(x_1, x_2, \dots, x_n\),

\[
\frac{x_1 + x_2 + \dots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \dots x_n}
\]

In our case, let's take \(x_i = 1 + a_i\) where \(a_i \geq 0\). Then we apply AM-GM:

\[
\frac{(1 + a_1) + (1 + a_2) + \dots + (1 + a_n)}{n} \geq \sqrt[n]{(1 + a_1)(1 + a_2) \dots (1 + a_n)}
\]

Simplifying the left-hand side:

\[
\frac{n + a_1 + a_2 + \dots + a_n}{n} \geq \sqrt[n]{(1 + a_1)(1 + a_2) \dots (1 + a_n)}
\]

This simplifies to:

\[
1 + \frac{a_1 + a_2 + \dots + a_n}{n} \geq \sqrt[n]{(1 + a_1)(1 + a_2) \dots (1 + a_n)}
\]

By the AM-GM inequality, this means:

\[
(1 + a_1)(1 + a_2)\dots(1 + a_n) \geq 2^n
\]

The inequality holds with equality if and only if \(a_1 = a_2 = \dots = a_n = 1\). 

Thus, for general non-negative \(a_i\), the inequality will always hold true, and if all \(a_i = 1\), the inequality becomes equality.

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Would you like further details or clarification on this result?

Here are 5 related questions to deepen your understanding:
1. What conditions lead to equality in the AM-GM inequality?
2. How does the AM-GM inequality apply to other algebraic problems?
3. Can you generalize this result to cases where \(a_i\) are negative?
4. How does this inequality relate to other inequalities like Cauchy-Schwarz?
5. What happens if we increase the number of terms \(n\) in this inequality?

**Tip:** The AM-GM inequality is a powerful tool for proving various inequalities and can often help simplify complex expressions into manageable forms.

Explore how to solve the inequality (1+a1)(1+a2)...(1+an) ≥ 2^n using the AM-GM inequality. This detailed solution walks through applying the Arithmetic Mean-Geometric Mean inequality to algebraic expressions, ideal for students in Grades 10-12.

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