Question-184503

Question Number 184503 by mr W last updated on 07/Jan/23

Answered by SEKRET last updated on 07/Jan/23

a = \sqrt{11 + a}

a^{2} = 11 + a

a^{3} = 11 a + a^{2} = 11 a + 11 + a = 12 a + 11

a^{4} = 12 a^{2} + 11 a = 12 \cdot 11 + 12 a + 11 a = 12 \cdot 11 + 23 a

a^{4} - a^{3} - 11 a^{2} + 11 =

12 \cdot 11 + 23 a - 12 a - 11 - 11^{2} - 11 a + 11 =

= 12 \cdot 11 - 11^{2} + 23 a - 23 a =

= 11 (12 - 11) = 1 \cdot 11 = 11

Commented by SEKRET last updated on 07/Jan/23

a = \sqrt{11 + \sqrt{11 + \sqrt{11 + \sqrt{11 + \sqrt{11 + \sqrt{11 + \sqrt{11 + 11 + \sqrt{11 + \dots .}}}}}}}}

Commented by mr W last updated on 07/Jan/23

a^{2} - a = 11

a^{4} - a^{3} - 11 a^{2} + 11

= a^{2} (a^{2} - a) - 11 a^{2} + 11

= 11 a^{2} - 11 a^{2} + 11

= 11

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