Question

Let $a_{1},a_{2},....,a_{n}$ be positive non-zero real numbers. If $a_{1},a_{2},....,a_{n}=k$, then the minimum value of $a_{1}+a_{2}+....+a_{n}$ is ________.

• A. $n(k)^{2/n}$
• B. $n(k)^{1/n}$
• C. $(k)^{1/n}$
• D. $(k)^{2/n}$
• E. $2n(k)^{2/n}$

Hint:

Sum is minimum when all numbers are equal ($AM=GM$).

Answer 1

The Correct Option is B

Step 1: Concept
Arithmetic Mean (AM) is always greater than or equal to Geometric Mean (GM) for positive numbers.

Step 2: Meaning
$AM = \frac{a_1 + a_2 + \dots + a_n}{n}$ and $GM = (a_1 \cdot a_2 \cdot \dots \cdot a_n)^{1/n}$.

Step 3: Analysis
Given $a_1 \cdot a_2 \cdot \dots \cdot a_n = k$, then $GM = k^{1/n}$. Using $AM \ge GM$: $\frac{a_1 + a_2 + \dots + a_n}{n} \ge k^{1/n}$.

Step 4: Conclusion
$a_1 + a_2 + \dots + a_n \ge n(k^{1/n})$. The minimum value is $n(k)^{1/n}$. Final Answer: (B)