$\lim_{n \to \infty} \frac{5 ^n}{\left(5-\sqrt{3}+2\sin x^\circ \right)^{n+2}} = \frac{1}{y}$

First split limit into: $\lim _{ n\rightarrow \infty }{ \frac { { 5 }^{ n } }{ { (5-\sqrt { 3 } +2\cdot \sin { (x) } ) }^{ n+2 } } } =\lim _{ n\rightarrow \infty }{ { (\frac { 5 }{ 5-\sqrt { 3 } +2\cdot \sin { (x } ) } ) }^{ n }\cdot \frac { 1 }{ { (5-\sqrt { 3 } +2\cdot \sin { (x) } ) }^{ 2 } } }$ $..$

Since limit must be expressed as $\frac { 1 }{ y }$ (y is real number), it can not be zero.
$..$
Now observe that factor on the left will not converge to 0 only if it's numerator equals it's denominator. $..$
Considering this, $\sin { (x) }$ must equal to $\frac { \sqrt { 3 } }{ 2 }$ therefore: $x={ 60 }^{ \circ }$ $..$
Limit then equals to $\frac { 1 }{ 25 }$ and $x+y=\boxed { 85 }$