The value of $$\left[ {1 + \frac{1}{{x + 1}}} \right]$$ $$\left[ {1 + \frac{1}{{x + 2}}} \right]$$ $$\left[ {1 + \frac{1}{{x + 3}}} \right]$$ $$\left[ {1 + \frac{1}{{x + 4}}} \right]$$ $${\text{is}} = ?$$
A. $$\frac{{x + 5}}{{x + 1}}$$
B. $$\frac{{x + 1}}{{x + 5}}$$
C. $$1 + \frac{1}{{x + 5}}$$
D. $$\frac{1}{{x + 5}}$$
Answer: Option A
Solution(By Examveda Team)
$$\left[ {1 + \frac{1}{{x + 1}}} \right]$$ $$\left[ {1 + \frac{1}{{x + 2}}} \right]$$ $$\,\left[ {1 + \frac{1}{{x + 3}}} \right]$$ $$\left[ {1 + \frac{1}{{x + 4}}} \right]$$$$ = \left[ {\frac{{\left( {x + 1} \right) + 1}}{{x + 1}}} \right]$$ $$\left[ {\frac{{\left( {x + 2} \right) + 1}}{{x + 2}}} \right]$$ $$\left[ {\frac{{\left( {x + 3} \right) + 1}}{{x + 3}}} \right]$$ $$\left[ {\frac{{\left( {x + 4} \right) + 1}}{{x + 4}}} \right]$$
$$\eqalign{ & = \left( {\frac{{x + 2}}{{x + 1}}} \right)\left( {\frac{{x + 3}}{{x + 2}}} \right)\left( {\frac{{x + 4}}{{x + 3}}} \right)\left( {\frac{{x + 5}}{{x + 4}}} \right) \cr & = \frac{{x + 5}}{{x + 1}} \cr} $$
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