If a₁, a₂, ..., aₙ are positive real numbers whose product is a fixed number c, then the minimum val

Q: Wait, why can't we just say $a_1 = a_2 = \dots = a_n$ and solve for $a_n$?

That is a common trap! If you assume all $a_i$ are equal, you are ignoring the coefficient '2' in the sum. The AM-GM inequality requires the terms being summed to be the same as the terms being multiplied. If you change the sum, you must change the terms in the product accordingly.

Q: Why does the product become $2c$?

Great question. We defined our terms as $x_1=a_1, \dots, x_{n-1}=a_{n-1}, x_n=2a_n$. When you multiply these, you get $a_1 \cdot a_2 \cdot \dots \cdot a_{n-1} \cdot (2a_n)$. Since $a_1 \cdot a_2 \cdot \dots \cdot a_n = c$, this simplifies to $2 \cdot (a_1 \cdot a_2 \cdot \dots \cdot a_n) = 2c$.

Q: Is the minimum value always achieved?

Yes, the AM-GM inequality guarantees that the minimum value is achieved when all terms are equal. In this case, $a_1 = a_2 = \dots = a_{n-1} = 2a_n$. You can solve for the specific values of $a_i$ if needed, but the inequality gives you the minimum sum directly.

Question

If $a_1, a_2, \dots, a_n$ are positive real numbers whose product is a fixed number $c$, then the minimum value of $a_1 + a_2 + \dots + a_{n-1} + 2a_n$ is

Options:

(A) $n(2c)^{1/n}$

(B) $(n+1)c^{1/n}$

(C) $2nc^{1/n}$

(D) $(n+1)(2c)^{1/n}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer. Today, we are going to peel back the layers of a classic optimization problem.

We are given $n$ positive real numbers $a_1, a_2, \dots, a_n$ whose product is a fixed constant $c$. Our mission is to minimize the sum $S = a_1 + a_2 + \dots + a_{n-1} + 2a_n$.

At first glance, this looks like a standard sum, but that coefficient '2' on the final term is a subtle, beautiful trap. It breaks the symmetry. If we were just minimizing $a_1 + a_2 + \dots + a_n$, the answer would be simple, but with that '2', we must be more strategic.

### The AM-GM Inequality: A Bridge Between Worlds

Whenever you see a problem involving a sum and a product, your mind should immediately jump to the Arithmetic Mean-Geometric Mean (AM-GM) inequality. It is the bridge between the additive and multiplicative worlds.

The theorem states that for any $k$ positive numbers $x_1, x_2, \dots, x_k$, the arithmetic mean is always greater than or equal to the geometric mean:

$$\frac{x_1 + x_2 + \dots + x_k}{k} \ge (x_1 \cdot x_2 \cdot \dots \cdot x_k)^{\frac{1}{k}}$$

The equality holds if and only if all the terms are equal. This is our most powerful tool for finding minimums.

### The Art of Strategic Grouping

Now, here is the masterclass moment. We need to apply AM-GM to our sum $S = a_1 + a_2 + \dots + a_{n-1} + 2a_n$.

If we just take the AM-GM of $a_1, a_2, \dots, a_n$, we get the product $c$, which is fine, but it does not account for the coefficient '2'. Instead, we treat the terms of our sum as the variables for the inequality.

Let our $n$ terms be $x_1 = a_1, x_2 = a_2, \dots, x_{n-1} = a_{n-1}$, and $x_n = 2a_n$. By grouping $2a_n$ as a single term, we have created a set of $n$ terms whose sum is exactly our objective $S$.

### The Final Synthesis

Now, let's calculate the product of these $n$ terms. The product $P = x_1 \cdot x_2 \cdot \dots \cdot x_n$ becomes:

$$P = a_1 \cdot a_2 \cdot \dots \cdot a_{n-1} \cdot (2a_n) = 2(a_1 \cdot a_2 \cdot \dots \cdot a_n)$$

Since we know the product of all $a_i$ is $c$, our product $P$ is simply $2c$. Now, we apply the AM-GM inequality:

$$\frac{a_1 + a_2 + \dots + a_{n-1} + 2a_n}{n} \ge (2c)^{\frac{1}{n}}$$

Multiplying both sides by $n$, we obtain the minimum value:

$$S \ge n(2c)^{\frac{1}{n}}$$

This is the **minimum value**. It is elegant, it is precise, and it shows how a simple change in perspective can unlock the solution. Keep practicing this, and you will start to see these patterns everywhere.

Animated Solution for Mathematics - Sequence and Series: If $a_{1}, a_{2}, \dots, a_{n}$ are positive real numbers whose product is a fixed number $c$ , then the minimum value of $a_{1} + a_{2} + \dots + a_{n - 1} + 2 a_{n}$ is

Select Answer:

Visualized Solution

Identifying the Constraints

The AM-GM Inequality Tool

Selecting the Terms

Calculating the Product

Applying the Inequality

Finding the Minimum Value

The Sigma Insight: Relation Between A.M., G.M., and H.M.

Analyzing the Setup

The AM-GM Inequality

The Art of Strategic Grouping

The Final Synthesis

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