find the value of √6+√6+√6+......infinity

1.	find the value of √6+√6+√6+......infinity
Answer» Let\xa0{tex}x = \\sqrt { 6 + \\sqrt { 6 + \\sqrt { 6 + \\dots } } }{/tex}\xa0......(i){tex}\\Rightarrow{/tex}\xa0x2 =\xa0{tex}( \\sqrt { 6 + \\sqrt { 6 + \\sqrt { 6 . . } } } ) ^ { 2 }{/tex}\xa0[Squaring both sides]{tex}\\Rightarrow{/tex}\xa0x2 =\xa0{tex}6 + \\sqrt { 6 + \\sqrt { 6 + \\sqrt { 6 + \\ldots } } }{/tex}{tex}\\Rightarrow{/tex}\xa0{tex}x^2 = 6 + x{/tex} [From (i)]{tex}\\Rightarrow{/tex}\xa0{tex}x^2 - x - 6 = 0{/tex}{tex}\\Rightarrow{/tex}\xa0{tex}(x - 3) (x + 2) = 0{/tex}{tex}\\Rightarrow{/tex}\xa0{tex}x = 3, x = -2{/tex}{tex}\\therefore{/tex}\xa0x = 3 [{tex}\\because{/tex}\xa0x > 0]

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