Embibe Experts Solutions for Chapter: Algebra, Exercise 4: EXERCISE - 1.4

If $a_{n+1}=\frac{1}{1+{\displaystyle \frac{1}{a_n}}}\left(n=1,2,\dots ,2008\right)$ and $a_1=1$, find the value of $a_1{a}_2+a_2{a}_3+a_3{a}_4+\dots +a_{2008}{a}_{2009}$

Given

$\frac{1}{x}+\frac{2}{y}+\frac{3}{z}=0 ...\left(\mathrm{i}\right)$

$\frac{1}{x}-\frac{6}{y}-\frac{5}{z}=0 ...\left(\mathrm{ii}\right)$

Find the absolute value of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$

Suppose that $f\left(x+3\right)=3x^2+7x+4$ and $f\left(x\right)=a{x}^2+bx+c$. What is $(a+b+c{)}^4$?

Let $a, b, c, d$, and $e$ be distinct integers such that $(6-a)(6-b)(6-c)(6-d)(6-e)=45$. What is $a+b+c+d+e$?

Let $m, n$ be integers such that $1<m\le n$. Define $f\left(m,n\right)=\left(1-\frac{1}{ m}\right)\times \left(1-\frac{1}{ m+1}\right)\times \left(1-\frac{1}{ m+2}\right)\times \dots \times \left(1-\frac{1}{n}\right).$

If $S=f(2,2049)+f(3,2049)+f(4,2049)+\dots +f(2049,2049)$, find the value of $\frac{S}{64}$

The number of positive integral solutions $(a,b,c,d)$ satisfying $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=1$ with the condition that $a<b<c<d$ is

For integers $a_1,\dots ,a_n\in \left\{1,2,3,\dots ,9\right\}$, we use the notation $\overline{a_1{a}_2\dots a_n}$ to denote the number ${10}^{n-1}{a}_1+{10}^{n-2}{a}_2+\dots +10a_{n-1}+a_n.$ For example, when $a=2$ and $b=0,\overline{ ab}$ denotes the number $20$. Given that $\overline{ab}=b^2$ and $\overline{acbc}=(\overline{ba}{)}^2$. Find the value of $a+b+c$.

It is known that there is only one pair of positive integers $a$ and $b$ such that $a\le b$ and $a^2+b^2+8ab=2010$. Find the value of $a+b$.