2-1-3-1-5-1-7-1-11-1-13-1-17-x-x-

Question Number 60577 by naka3546 last updated on 22/May/19

2 \sqrt{1 + 3 \sqrt{1 + 5 \sqrt{1 + 7 \sqrt{1 + 11 \sqrt{1 + 13 \sqrt{1 + 17 \sqrt{\dots}}}}}}} = x

x = ?

Commented by ajfour last updated on 22/May/19

Is there such a question in a book ?

Answered by Prithwish sen last updated on 22/May/19

(n + 1) (n + 3) = (n + 1) \sqrt{{(n + 3)}^{2}}

= (n + 1) \sqrt{1 + (n + 2) (n + 4)}

= (n + 1) \sqrt{1 + (n + 2) \sqrt{{(n + 4)}^{2}}}

= (n + 1) \sqrt{1 + (n + 2) \sqrt{1 + (n + 3) (n + 5)}}

= (n + 1) \sqrt{1 + (n + 2) \sqrt{1 + (n + 3) \sqrt{{(n + 5)}^{2}}}}

and so on

putting n = l

8 = 2 \sqrt{1 + 3 \sqrt{1 + 4 \sqrt{1 + 5 \sqrt{1 + \dots \dots .}}}}

is the qiestion ok ?

Commented by ajfour last updated on 22/May/19

yes Sir, very fine one this is .

Commented by Prithwish sen last updated on 22/May/19

thank you .