real analysis - 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. ${1+\sqrt{33}}\over{2}$




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




3. ${1+\sqrt{30}}\over{2}$




4. ${1+\sqrt{29}}\over{2}$

How to proceed now.
Thanks for any help.

Answer

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.