If {a_1}, {a_2}, ..., {a_n} are positive real | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) By the $AM-GM$ inequality,for $n$ positive real numbers,the arithmetic mean is greater than or equal to the geometric mean.Consider the $n$ number

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) By the $AM-GM$ inequality,for $n$ positive real numbers,the arithmetic mean is greater than or equal to the geometric mean.Consider the $n$ numbers: ${a_1}, {a_2}, ..., {a_{n-1}}, 2{a_n}$.Their arithmetic mean is $\frac{{a_1} + {a_2} + ... + {a_{n-1}} + 2{a_n}}{n}$.Their geometric mean is $({a_1} \cdot {a_2} \cdot ... \cdot {a_{n-1}} \cdot 2{a_n})^{1/n} = (2 \cdot {a_1} \cdot {a_2} \cdot ... \cdot {a_n})^{1/n} = (2c)^{1/n}$.Applying $AM \ge GM$:$\frac{{a_1} + {a_2} + ... + {a_{n-1}} + 2{a_n}}{n} \ge (2c)^{1/n}$.Therefore,the minimum value of ${a_1} + {a_2} + ... + {a_{n-1}} + 2{a_n}$ is $n(2c)^{1/n}$.

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

Similar Questions

If $\log _{5} 2, \log _{5}(2^{x}-3)$ and $\log _{5}(\frac{17}{2}+2^{x-1})$ are in $A.P.$,then the value of $x$ is:

The sum of the series $\frac{1}{2} + \frac{1}{3} + \frac{1}{6} + \dots$ to $9$ terms is

Let $a_1, a_2, a_3, \ldots$ be in an $A.P.$ such that $\sum_{k=1}^{12} a_{2k-1} = -\frac{72}{5} a_1$,where $a_1 \neq 0$. If $\sum_{k=1}^{n} a_k = 0$,then $n$ is:

The $20^{\text{th}}$ term from the end of the progression $20, 19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots, -129 \frac{1}{4}$ is:

The number of terms in an $A.P.$ is even. The sum of the odd terms is $24$ and the sum of the even terms is $30$. If the last term exceeds the first term by $10\frac{1}{2}$,then the number of terms in the $A.P.$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module