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

Answered by SEKRET last updated on 07/Jan/23

$$\mathbf{a}=\sqrt{11+\mathbf{a}}$$$${\mathbf{a}}^2=11+\mathbf{a}$$$${\mathbf{a}}^3=11\mathbf{a}+{\mathbf{a}}^2=11\mathbf{a}+11+\mathbf{a}=12\mathbf{a}+11$$$${\mathbf{a}}^4=12{\mathbf{a}}^2+11\mathbf{a}=12\cdot 11+12\mathbf{a}+11\mathbf{a}=12\cdot 11+23\mathbf{a}$$$${\mathbf{a}}^4-{\mathbf{a}}^3-11{\mathbf{a}}^2+11=$$$$12\cdot 11+23\mathbf{a}-12\mathbf{a}-11-{11}^2-11\mathbf{a}+11=$$$$=12\cdot 11-{11}^2+23\mathbf{a}-23\mathbf{a}=$$$$=11(12-11)=1\cdot 11=11$$

Commented by SEKRET last updated on 07/Jan/23

$$\mathbf{a}=\sqrt{11+\sqrt{11+\sqrt{11+\sqrt{11+\sqrt{11+\sqrt{11+\sqrt{11+11+\sqrt{11+....}}}}}}}}$$

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

$$a^2-a=11$$$$a^4-a^3-11a^2+11$$$$=a^2(a^2-a)-11a^2+11$$$$=11a^2-11a^2+11$$$$=11$$

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