Let n geq 4 be a positive integer and let l_1 | vedclass

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

Question

(D) Given $\frac{l_1}{l_2} + \frac{l_2}{l_3} + \ldots + \frac{l_n}{l_1} = n$.By the Arithmetic Mean-Geometric Mean $(AM \geq GM)$ inequality,we have $

Vedclass · Accepted Answer

$I$ and $III$ are true

Vedclass · Answer

(D) Given $\frac{l_1}{l_2} + \frac{l_2}{l_3} + \ldots + \frac{l_n}{l_1} = n$.By the Arithmetic Mean-Geometric Mean $(AM \geq GM)$ inequality,we have $\frac{1}{n} \sum_{i=1}^{n} \frac{l_i}{l_{i+1}} \geq \sqrt[n]{\prod_{i=1}^{n} \frac{l_i}{l_{i+1}}}$,where $l_{n+1} = l_1$.Since the product of the terms is $\frac{l_1}{l_2} \cdot \frac{l_2}{l_3} \cdots \frac{l_n}{l_1} = 1$,the $GM$ is $1$.Thus,$\frac{1}{n} \sum \frac{l_i}{l_{i+1}} \geq 1$,which implies $\sum \frac{l_i}{l_{i+1}} \geq n$.The equality holds if and only if all terms are equal,i.e.,$\frac{l_1}{l_2} = \frac{l_2}{l_3} = \ldots = \frac{l_n}{l_1} = 1$.This implies $l_1 = l_2 = \ldots = l_n$,so statement $I$ is true.If $P$ is cyclic and all sides are equal,then the arcs subtended by these sides are equal,which implies the angles are equal. Thus,$P$ is a regular polygon,making statement $III$ true.Statement $II$ is not necessarily true because equal sides do not imply equal angles in a general polygon (e.g.,a rhombus is not necessarily a square).

Similar Questions

Two sequences $\{t_n\}$ and $\{s_n\}$ are defined by $t_n = \log \left( \frac{5^{n+1}}{3^{n-1}} \right)$ and $s_n = \left[ \log \left( \frac{5}{3} \right) \right]^n$. Then:

Let $\alpha = \sum_{n=101}^{200} 2^n \sum_{k=101}^n \frac{1}{k !}$ and $b = \sum_{n=101}^{200} \frac{2^{201}-2^n}{n !}$. Then,$\frac{a}{b}$ is

Let $a_0=0$ and $a_n=3 a_{n-1}+1$ for $n \geq 1$. Then,the remainder obtained by dividing $a_{2010}$ by $11$ is

If all the terms of an $A.P.$ are squared,then the new series will be in

Let $a_1, a_2, a_3, \ldots$ be an $A$.$P$. If $a_7 = 3$,the product $a_1 a_4$ is minimum and the sum of its first $n$ terms is zero,then $n! - 4 a_{n(n+2)}$ is equal to :

Vedclass Test Series

Exam Paper Generator

Online Exam Module