已知数列满足a1,b1＞0 a(n+1)=an+1/bn
b(n+1)=bn+1/an 求证a50+b50＞20
求大神

a(n+1)
=a(n)+( 1 / b(n) )
=( a(n)*b(n) + 1 ) / b(n)
=a(n) * { ( a(n)*b(n) + 1 ) / ( a(n)*b(n) ) }
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b(n+1)
=b(n)+( 1 / a(n) )
=( a(n)*b(n) + 1 ) / a(n)
=b(n) * { ( a(n)*b(n) + 1 ) / ( a(n)*b(n) ) }
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故 a(k)/b(k)為定值

设a(n)/a(1)=c(n)，则：
c(n+1)
=a(n+1)/a(1)
=[a(n) * { ( a(n)*b(n) + 1 ) / ( a(n)*b(n) ) } ]/ a(1)
=a(n)/a(1) * { ( a(n)*b(n) + 1 ) / ( a(n)*b(n) ) }
=c(n) * { ( a(n)*b(n) + 1 ) / ( a(n)*b(n) ) }

又：c(n)=a(n)/a(1)； 又：a(n)/b(n)=a(1)/b(1)
故：c(n)=a(n)/a(1)＝b(n)/b(1)
故：a(n)＝a(1)*c(n)；b(n)＝b(1)*c(n)
又：c(n+1)
=c(n) * { ( a(n)*b(n) + 1 ) / ( a(n)*b(n) ) }
（代入得）
=c(n) * { ( a(1)*c(n)*b(1)*c(n) + 1 ) / ( a(1)*c(n)*b(1)*c(n) ) }
=( a(1)*b(1)*c(n)*c(n) + 1 ) / ( a(1)*b(1)*c(n) )

令D(n)=c(n) * √(a(1)*b(1))，则c(n)=D(n) / √(a(1)*b(1))
则：D(n+1)
=c(n+1) * √(a(1)*b(1))
={ ( a(1)*b(1)*c(n)*c(n) + 1 ) / ( a(1)*b(1)*c(n) ) } * √(a(1) *b(1))
={c(n)+ 1 / ( a(1)*b(1)*c(n) )} * √(a(1)*b(1))
={D(n)*/√(a(1)*b(1))+ 1 / ( a(1)*b(1)*D(n)/√(a(1)*b(1) ))} * √(a(1)*b(1))
={D(n)*/√(a(1)*b(1))+ 1 / ( √(a(1)*b(1)))*D(n)}) * √(a(1 )*b(1))
=D(n) + 1/D(n)

则：a(n)+b(n)
= a(1)*c(n) +b(1)*c(n)
=( a(1) + b(1) ) * c(n)
=( a(1) + b(1) ) * D(n)/√(a(1)*b(1))
=( a(1) + b(1) )/√(a(1)*b(1)) * D(n)
因为a(1) + b(1) - 2√(a(1)*b(1))=( √(a(1)) - √(b(1)) )^2 >= 0，所以上式：
>=2 * D(n)

又：D(2) - 2
=D(1) + 1/D(1) - 2
=(√D(1) - 1/√D(1))^2
>=0
故：D(2) >= 2

令f(x)=x+1/x
则：f'(x)=1 - 1/(x^2)
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当 x>1 时，f'(x)=1 - 1/(x^2)恒大于零
故：当 x>1 时，f(x)递增
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令K(2)=2，且K(n+1) = f( K(n) ) = K(n)+1/K(n)
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倘若D(r)>=K(r)>=1，则：
因当 x>1 时，f(x)递增：
故：f( D(r) )>= f( K(r) )
故：D(r+1)>= K(r+1)
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又：D(2) >= 2 =K(2)
故由数学归纳法知：若n属于大于2的正整数，则D(n) 恒大于等于K(n)

n: K(n)
1: 2
2: 2.5
3: 2.9
4: 3.24483
5: 3.55301
6: 3.83446
7: 4.09525
8: 4.33944
9: 4.56988
10: 4.78871
11: 4.99753
12: 5.19763
13: 5.39003
14: 5.57555
15: 5.75491
16: 5.92867
17: 6.09735
18: 6.26135
19: 6.42106
20: 6.5768
21: 6.72885
22: 6.87746
23: 7.02286
24: 7.16526
25: 7.30482
26: 7.44171
27: 7.57609
28: 7.70809
29: 7.83782
30: 7.96541
31: 8.09095
32: 8.21454
33: 8.33628
34: 8.45624
35: 8.57449
36: 8.69112
37: 8.80618
38: 8.91974
39: 9.03185
40: 9.14257
41: 9.25194
42: 9.36003
43: 9.46687
44: 9.5725
45: 9.67696
46: 9.7803
47: 9.88255
48: 9.98374
49: 10.0839
50: 10.1831

因a(n)+b(n) >= 2*D(n) >= 2*K(n)
又K(50)>10
故a(50)+b(50)>2*10=20
＃＃＃故得证＃＃＃

对于本题有些步骤没必要
设C(n)=a(n)+b(n),于是C(2)=a1+1/a1+a2+1/a2>=4
而且C(n+1)=C(n)+1/a(n)+1/b(n)>=C(n)+4/C(n)
于是定义数列
K_2=4,K_{n+1}=d_n+4/d_n,必然有C_n>=K_n

K_2=4,K_{n+1}=K_n+4/K_n,必然有C_n>=K_n
K_{n+1}^2=k_n^2+8+16/K_n^2>=k_n^2+8
所以
K_50^2>=8*48+k_2^2=400
得出C_50>=K_50>=20