Embibe Experts Solutions for Chapter: Sequences and Series, Exercise 2: EXERCISE-2

Product of $n$ positive numbers is unity. The sum of these numbers can not be less than-

Let $a_1,a_2,a_3\dots \dots$ and $b_1,b_2,b_3,\dots \dots$ be arithmetic progression such that $a_1=25,b_1=75$ and $a_{100}+b_{100}=100$, then 

If the AM of two positive numbers be three times their geometric mean then the ratio of the numbers is

If the first and ${\left(2n-1\right)}^{\mathrm{th}}$ terms of an A.P., a G.P. and a H.P. are equal and their $n^{\mathrm{th}}$ terms are $a,b$ and $c$ respectively, then

The $p^{th }$ term $T_p$ of $H.P.$ is$q(q+p)$ and $q^{th}$ term $T_q$ is $p(p+q)$ when $p>1, q>1$, then

$a, b, c$ are three distinct real numbers, which are in $G.P.$ and $a+b+c=x b$, then -

If $10$ harmonic means $H_1,H_2,H_3\dots \dots .H_{10}$ are inserted between $7$ and $-\frac{1}{3}$, then -

Indicate the correct alternative(s), for $0<\varphi <\frac{\pi }{2}$, if $x=\sum _{n=0}^{\infty }{\cos }^{2n}\varphi ,y=\sum _{n=0}^{\infty }{\sin }^{2n}\varphi$ and $z=\sum _{n=0}^{\infty }{\cos }^{2n}\varphi {\sin }^{2n}\varphi$. then