question_answer

If \[m=\sqrt{5+\sqrt{5+\sqrt{5+.....}}}\] and \[n=\sqrt{5-\sqrt{5-\sqrt{5-.....}}},\] then among the following the relation between m and n holds is                                                                                                 [SSC (CGL) Mains 2014]

A) \[m+n-1=0\]                 

B) \[m-n-1=0\]

C)                 

D)

Correct Answer: B

Solution :

\[m=\sqrt{5+\sqrt{5+\sqrt{5+....}}}\]

\[\Rightarrow \]   \[{{m}^{2}}=5+m\]    [both side squaring]

\[\Rightarrow \]\[{{m}^{2}}-m-5=0\]                              ... (i)

\[n=\sqrt{5-\sqrt{5-\sqrt{5-....}}}\]

\[\Rightarrow \]   \[{{n}^{2}}=5-n\]   [again, both side squaring]

\[\Rightarrow \]   \[{{n}^{2}}+n-5=0\]                           ... (ii)

On subtracting Eq. (i) from Eq, (ii), we get

\[{{m}^{2}}-m-{{n}^{2}}-n=0\]

\[\Rightarrow \]               \[{{m}^{2}}-{{n}^{2}}-m-n=0\]

\[\Rightarrow \] \[(m+n)(m-n)-(m+n)=0\]

\[\Rightarrow \]   \[(m+n)(m-n-1)=0\]

\[\because \]       \[m+n\ne 0\]

\[\therefore \]      x