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

The sum of the first four terms of an $A.P.$ is $56$. The sum of the last four terms is $112$. If its first term is $11$,then find the number of terms.

The value of $\sum\limits_{r = 1}^n {\log \left( {\frac{{{a^r}}}{{{b^{r - 1}}}}} \right)} $ is

If $a, b,$ and $c$ are positive real numbers,then the minimum value of $(a + b + c)(1/a + 1/b + 1/c)$ is .......

Let $a_1=8, a_2, a_3, \ldots, a_n$ be an $A.P.$ If the sum of its first four terms is $50$ and the sum of its last four terms is $170$,then the product of its middle two terms is

If the sum of $n$ terms of an $A.P.$ is $nA + n^2B$,where $A$ and $B$ are constants,then its common difference will be

Vedclass Test Series

Exam Paper Generator

Online Exam Module