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

The sum of all the elements in the set $\{n\in \{1,2,\dots \dots ,100\}\mid$ H.C.F. of $n$ and $2040$ is $1$} is equal to __________.

If the first, second and last term of an arithmetic series are $a,b,c$ respectively, them the number of terms is

If $a,b,c$ are in arithmetic progression, then the value of $\left(a+2b-c\right)\left(2b+c-a\right)\left(a+2b+c\right)$ is

If in a G.P., the ratio of the sum of the first eleven terms to the sum of the last eleven terms is $\frac{1}{8}$ and the ratio of the sum of all the terms without the first nine to the sum of all the terms without the last nine is $2$, then the number of terms in the G.P is:

The series $\frac{2x}{x+3}+{\left(\frac{2x}{x+3}\right)}^2+{\left(\frac{2x}{x+3}\right)}^3+\dots \dots \infty$ will have a definite sum when -

Let $a, b, c, d$ and $e$ be distinct positive numbers. If $a, b, c$ and $\frac{1}{c}$, $\frac{1}{d}$, $\frac{1}{e}$ both are in $\mathrm{AP}$ and $b, c, d$ are in $\mathrm{GP}$, then

If three distinct positive numbers $a, b$ and $c$ are in $\mathrm{AP}$ such that $abc=4$, then the value of $b$ is always

Let $m$ be the smallest odd positive integer for which $1+2+\dots \dots +m$ is a square of an integer, and let $n$ be the smallest even positive integer for which $1+2+\dots \dots +n$ is a square of an integer. What is the value of $m+n?$