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

If $x=\sum _{n=0}^{\infty }{a}^n,y=\sum _{n=0}^{\infty }{b}^n,z=\sum _{n=0}^{\infty }{c}^n$, where $a,b,c$ are in $\mathrm{AP}$ and $\left|a\right|<1,\left|b\right|<1,\left|c\right|<1$, then $x,y$ and $z$ are in

If $a_1, a_2,\dots ,a_n$ are positive real numbers whose product is a fixed number $c$, then the minimum value of $a_1+a_2+\dots .+a_{n-1}+2a_n$ is

The sum of the first $n$ terms of the series $1^2+2·2^2+3^2+2·4^2+5^2+2·6^2+\dots$ is $\frac{n{\left(n+1\right)}^2}{2}$ where $n$ is even. When $n$ is odd, then the sum is

Let $T,$ and $S,$ be the $r^{\text{th }}$ term and sum up to $r^{\text{th}}$ term of a series respectively. If for an odd number $n, S_n=n$ and $T_n=\frac{T_{n-1}}{n^2},$ then $T_m (m$ being even$)$ is

A man arranges to pay off a debt of $\mathrm{Rs}. 3600$ by $40$ annual installments which form an arithmetic series, when $30$ of the installments are paid he dies leaving a third of the debt unpaid, find the value of the first installment

The number of terms in an $\mathrm{A}.\mathrm{P}.$ is even, the sum of the odd terms is $24$, of the even terms $30$, and the last term exceeds the first term by $10\frac{1}{2}$, find the number of terms.

Consider an infinite geometric series with first term $a$ and common ratio $r$. If the sum is $4$ and the second term is $\frac{3}{4}$, then

The roots of the equation $x^4-8x^3+a{x}^2-bx+16=0,$ are positive, if