If a_1, a_2, dots, a_n are positive real numb | vedclass

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

Question

(C) For positive real numbers $a_1, a_2, \dots, a_n$,the Arithmetic Mean-Geometric Mean $(AM-GM)$ inequality states that $AM \ge GM$.$	herefore \frac

Vedclass · Accepted Answer

Not less than $n$.

Vedclass · Answer

(C) For positive real numbers $a_1, a_2, \dots, a_n$,the Arithmetic Mean-Geometric Mean $(AM-GM)$ inequality states that $AM \ge GM$.$	herefore \frac{a_1 + a_2 + \dots + a_n}{n} \ge (a_1 \cdot a_2 \cdot \dots \cdot a_n)^{\frac{1}{n}}$Given $a_1 \cdot a_2 \cdot \dots \cdot a_n = 1$,we substitute this into the inequality:$	herefore \frac{a_1 + a_2 + \dots + a_n}{n} \ge (1)^{\frac{1}{n}}$$	herefore \frac{a_1 + a_2 + \dots + a_n}{n} \ge 1$$	herefore a_1 + a_2 + \dots + a_n \ge n$Thus,the sum of the numbers is not less than $n$.

If $a_1, a_2, \dots, a_n$ are positive real numbers such that $a_1 \cdot a_2 \cdot \dots \cdot a_n = 1$,then their sum is:

Similar Questions

The sum of two numbers is $6$ times their geometric mean. Show that the numbers are in the ratio $(3+2 \sqrt{2}):(3-2 \sqrt{2})$.

If $a, b, c$ are in $A.P.$ and $a, c - b, b - a$ are in $G.P.$ $(a \ne b \ne c)$,then $a:b:c$ is

If $A$,$G$,and $H$ are the arithmetic,geometric,and harmonic means between two positive real numbers,respectively,then:

If the geometric mean between $a$ and $b$ is $\frac{a^{n+1} + b^{n+1}}{a^n + b^n}$,then what is the value of $n$?

If the ratio of the Arithmetic Mean and Harmonic Mean of two positive real numbers $a$ and $b$ is $m:n$,then find the value of $a:b$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module