직각삼각형을 반복하여 만든 수열의 극한

$\displaystyle \overline{\mathrm{P} _{1} P _{2}} = 1$이고 $\displaystyle \overline{{\mathrm{P}} _{n} {\mathrm{P}} _{n+1}} = 2 \overline{{\mathrm{P}} _{n-1} {\mathrm{P}} _{n}}$이므로 $\displaystyle \overline{{\mathrm{P}} _{n} {\mathrm{P}} _{n+1}} = 2 ^{n-1}$ $\displaystyle \therefore {\overline{{\mathrm{P}} _{n} {\mathrm{P}} _{n+1}}} ^{2} = 4 ^{n-1}$ $\displaystyle {\overline{{\mathrm{AP}} _{n+1}}} ^{2}$$\displaystyle = {\overline{{\mathrm{AP}} _{n}}} ^{2} + {\overline{{\mathrm{P}} _{n} {\mathrm{P}} _{n+1}}} ^{2}$$\displaystyle = {\overline{{\mathrm{AP}} _{n}}} ^{2} + 4 ^{n-1}$ $\displaystyle \therefore {\overline{{\mathrm{AP}} _{n}}} ^{2}$$\displaystyle = \overline{\mathrm{AP} _{1}} + \sum\limits _{k=1} ^{n-1} 4 ^{k-1}$ $\displaystyle = 1 + \frac{4 ^{n-1} - 1}{4 - 1}$$\displaystyle = \frac{4 ^{n-1}}{3} + \frac{2}{3}$ $\displaystyle \therefore \lim\limits _{n\rightarrow \infty} \left( \frac{\overline{{\mathrm{P}} _{n} {\mathrm{P}} _{n+1}}}{\overline{{\mathrm{AP}} _{n}}} \right) ^{2}$$\displaystyle = \lim\limits _{n\rightarrow \infty} \frac{4 ^{n-1}}{\frac{4 ^{n-1}}{3} + \frac{2}{3}}$$= 3$