数学のブログ

数と極限 数のいろいろ、漸化式 フィボナッチ数、一般項、数学的帰納法

微分積分演習〈理工系の数学入門コース/演習 新装版〉 (和達 三樹(著)、十河 清(著)、岩波書店)の第1章(数と極限)、1-1(数のいろいろ、漸化式)、問題2の解答を求めてみる。

1 5 ( 1 + 5 2 - 1 - 5 2 ) = 1
1 5 ( ( 1 + 5 2 ) 2 - ( 1 - 5 2 ) 2 )
= 1 4 5 · 4 5
= 1

また、

F n = F n - 1 + F n - 2
= 1 5 ( ( 1 + 5 2 ) n - ( 1 - 5 2 ) n ) + 1 5 ( ( 1 + 5 2 ) n - 1 - ( 1 - 5 2 ) n - 1 )
= 1 5 ( ( 1 + 5 2 ) n - 1 ( 1 + 5 2 + 1 ) - ( 1 - 5 2 ) n - 1 ( 1 - 5 2 + 1 ) )
= 1 5 ( ( 1 + 5 2 ) n - 1 3 + 5 2 - ( 1 - 5 2 ) n - 1 3 - 5 2 )
= 1 5 ( ( 1 + 5 2 ) n - 1 6 + 2 5 4 - ( 1 - 5 2 ) n - 1 6 - 2 5 4 )
= 1 5 ( ( 1 + 5 2 ) n - 1 ( 1 + 5 2 ) 2 - ( 1 - 5 2 ) n - 1 ( 1 - 5 2 ) 2 )
= 1 5 ( ( 1 + 5 2 ) n + 1 - ( 1 - 5 2 ) n + 1 )

よって帰納法により成り立つ。

（証明終）

コード(Wolfram Language, Jupyter)

f[n_] := 1 / Sqrt[5] (((1+Sqrt[5])/2)^(n+1) - ((1-Sqrt[5])/2)^(n+1))

f[0]
Output
Simplify[%]

1
f[1] // Simplify

1
f[n] == f[n-1] + f[n-2]
Output
Simplify[%]
Output
Table[f[n], {n, 1, 100}]
Output
Simplify[%]

{1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 
 
>   10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 
 
>   2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 
 
>   102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 
 
>   2971215073, 4807526976, 7778742049, 12586269025, 20365011074, 32951280099, 
 
>   53316291173, 86267571272, 139583862445, 225851433717, 365435296162, 591286729879, 
 
>   956722026041, 1548008755920, 2504730781961, 4052739537881, 6557470319842, 
 
>   10610209857723, 17167680177565, 27777890035288, 44945570212853, 72723460248141, 
 
>   117669030460994, 190392490709135, 308061521170129, 498454011879264, 806515533049393, 
 
>   1304969544928657, 2111485077978050, 3416454622906707, 5527939700884757, 
 
>   8944394323791464, 14472334024676221, 23416728348467685, 37889062373143906, 
 
>   61305790721611591, 99194853094755497, 160500643816367088, 259695496911122585, 
 
>   420196140727489673, 679891637638612258, 1100087778366101931, 1779979416004714189, 
 
>   2880067194370816120, 4660046610375530309, 7540113804746346429, 12200160415121876738, 
 
>   19740274219868223167, 31940434634990099905, 51680708854858323072, 
 
>   83621143489848422977, 135301852344706746049, 218922995834555169026, 
 
>   354224848179261915075, 573147844013817084101}
ListLinePlot[%]
Output
ListLinePlot[Table[f[n], {n, 0, 10}]]
Output