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a(n+2)/an=[a(n+1)^2+1]/[an^2+1]
a(n+1)/a(n-1)=[an^2+1]/[a(n-1)^2+1]
an/a(n-2)=[a(n-1)^2+1]/[a(n-2)^2+1]
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a3/a1=(a2^2+1)/(a1^2+1)
左边右边分别相乘得到：
[a(n+2)*a(n+1)*(an)*……*a4*a3]/[an*a(n-1)*a(n-2)*……*a2*a1]
=[a(n+1)^2+1]/[a1^2+1]
→a(n+2)*a(n+1)/(a1a2)=[a(n+1)^2+1]/[a1^2+1]
→a(n+2)=[a(n+1)^2+1]/a(n+1)
→a(n+1)=[an^2+1]/an
a(n+1)/a(n-1)=[an^2+1]/[a(n-1)^2+1]
an/a(n-2)=[a(n-1)^2+1]/[a(n-2)^2+1]
...
...
...
a3/a1=(a2^2+1)/(a1^2+1)
左边右边分别相乘得到：
[a(n+2)*a(n+1)*(an)*……*a4*a3]/[an*a(n-1)*a(n-2)*……*a2*a1]
=[a(n+1)^2+1]/[a1^2+1]
→a(n+2)*a(n+1)/(a1a2)=[a(n+1)^2+1]/[a1^2+1]
→a(n+2)=[a(n+1)^2+1]/a(n+1)
→a(n+1)=[an^2+1]/an
