Limits with square root

Calculus Level 2

Find the limit value of the following sequence: 7 , 7 + 6 7 , 7 + 6 7 + 6 7 , ⋯ . \sqrt{7}, \sqrt{7+6\sqrt{7}}, \sqrt{7+6\sqrt{7+6\sqrt{7}}}, \cdots.

6 8 5 7

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Mahindra Jain by Mahindra Jain

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1 solution

Tunk-Fey Ariawan
Mar 11, 2014

Let x = 7 + 6 7 + 6 7 + ⋯ x=\sqrt{7+6\sqrt{7+6\sqrt{7+\cdots}}} , where x > 0 x>0 . Therefore

x 2 = 7 + 6 7 + 6 7 + ⋯ x 2 = 7 + 6 x x 2 − 6 x − 7 = 0 ( x + 1 ) ( x − 7 ) = 0. \begin{aligned} x^2&=7+6\sqrt{7+6\sqrt{7+\cdots}}\\ x^2&=7+6x\\ x^2-6x-7&=0\\ (x+1)(x-7)&=0. \end{aligned}

Thus, the possible solution is x = 7 x=\boxed{7} .

21 Helpful 0 Interesting 0 Brilliant 0 Confused

how did you do that O_O?

Mhykl Krysty - 7 years, 2 months ago

absolutely...

Max B - 7 years, 2 months ago

Cool. . .

Sundhara Moorthi - 7 years, 1 month ago

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