Where does this sequence $\sqrt{7}$,$\sqrt{7+ \sqrt{7}}$,$\sqrt{7+\sqrt{7+\sqrt{7}}}$,.... converge? The  given  sequence  is $\sqrt{7}$,$\sqrt{7+ \sqrt{7}}$,$\sqrt{7+\sqrt{7+\sqrt{7}}}$,.....and  so  on. 
the  sequence  is  increasing  so  to  converge  must  be  bounded  above.Now  looks  like  they  would not  exceed  7. The  given  options  are 


*

*${1+\sqrt{33}}\over{2}$

*${1+\sqrt{32}}\over{2}$

*${1+\sqrt{30}}\over{2}$

*${1+\sqrt{29}}\over{2}$
How  to  proceed now. 
Thanks  for  any  help.

Trick: Let $X = \sqrt{ 7 + \sqrt{ 7 + ... } } $. We have $X = \sqrt{ 7 + X } $ and so $X^2 = 7 + X $. Now you solve the quadratic equation.
