Find the value of𝑥 = √15+√15+√15+⋯

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Accepted Answer

1. Let $x=\sqrt{15+\sqrt{15+\cdots}}$. Then we have $x=\sqrt{15+x}$.  
2. Square both sides: $x^2=15+x$.  
3. Rearrange: $x^2-x-15=0$. Solve: $x=\dfrac{1\pm\sqrt{1+60}}{2}=\dfrac{1\pm\sqrt{61}}{2}$.  
4. Take the positive root (square roots are nonnegative): $x=\dfrac{1+\sqrt{61}}{2}$, which matches option (b).  
FINAL_ANSWER: b

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