The value of $(1 + \frac{1}{x}) (1 + \frac{1}{x + 1})$ $\left( {{\text{1 + }}\frac{1}{x}} \right)\left( {{\text{1 + }}\frac{1}{{x + 1}}} \right)$ $(1 + \frac{1}{x + 2}) (1 + \frac{1}{x + 3})$ $\left( {{\text{1 + }}\frac{1}{{x + 2}}} \right)\left( {{\text{1 + }}\frac{1}{{x + 3}}} \right)$ is?

1 $1 + \frac{1}{x + 4}$ $1 + \frac{1}{{x + 4}}$
2x + 4
3 $\frac{1}{x}$ $\frac{1}{x}$
4 $\frac{x + 4}{x}$ $\frac{{x + 4}}{x}$

Answer:- 1

Explanation:-

Solution:

\begin{aligned} (1 + \frac{1}{x}) (1 + \frac{1}{x + 1}) (1 + \frac{1}{x + 2}) (1 + \frac{1}{x + 3}) \\ Taking L .C .M of each term . \\ \Rightarrow (\frac{x + 1}{x}) (\frac{x + 1 + 1}{x + 1}) (\frac{x + 2 + 1}{x + 2}) (\frac{x + 3 + 1}{x + 3}) \\ \Rightarrow \frac{1}{x} \times (x + 4) \\ \Rightarrow \frac{x + 4}{x} \end{aligned}

$\eqalign{ & \left( {{\text{1 + }}\frac{1}{x}} \right)\left( {{\text{1 + }}\frac{1}{{x + 1}}} \right)\left( {{\text{1 + }}\frac{1}{{x + 2}}} \right)\left( {{\text{1 + }}\frac{1}{{x + 3}}} \right) \cr & {\text{Taking L}}{\text{.C}}{\text{.M of each term}}{\text{.}} \cr & \Rightarrow \left( {\frac{{x + 1}}{x}} \right)\left( {\frac{{x + 1 + 1}}{{x + 1}}} \right)\left( {\frac{{x + 2 + 1}}{{x + 2}}} \right)\left( {\frac{{x + 3 + 1}}{{x + 3}}} \right) \cr & \Rightarrow \frac{1}{x} \times \left( {x + 4} \right) \cr & \Rightarrow \frac{{x + 4}}{x} \cr}$