Is there a general formula for Fibonacci sequence

Is there a general formula for Fibonacci sequence

An = 1 / radical 5 {[(1 + radical 5) / 2] to the nth power - [(1 - radical 5) / 2] to the nth power} (n belongs to a positive integer). This sequence was proposed by Fibonacci in Italy in the 13th century, so it is called Fibonacci sequence. This sequence is determined by the following recurrence relation: F0 = 0, F1 = 1, FN + 2 = FN + FN + 1 (n > = 0)

Who will prove the general term formula of Fibonacci sequence by mathematical induction
Proving general term formula by mathematical induction

The general term formula of the sequence can be obtained by the undetermined coefficient method

Proving a general formula an = n by mathematical induction
Let's guess the general term formula of a sequence and prove it, where 3 (a1 + A2 +... + an) = (2n + 1) (a1 + A2 + a3 +... + an)
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When n = 1, 3A1 & # 178; = 3A1, A1 = 0 or 1 & nbsp; 0 are rounded off
If n = 2 is 3 (A1 & # 178; + A2 & # 178;) = 5 (a1 + A2), that is, 3 (1 & # 178; + A2 & # 178;) = 5 (1 + A2); 3a2 & # 178; - 5a2-2 = 0, then (A2-2) * (3a2 + 1) = 0, A2 = 2
Suppose n = k, that is, 3 (A1 & # 178; + A2 & # 178; +...) When AK & # 178;) = (2k + 1) (a1 + A2 + a3 +... + an), AK = k holds
Then when n = K + 1, 3 (1 & # 178; + 2 & # 178; +...) +k²+（ak+1）²）=(2(k+1)+1)(1+2+3…… k+（ak+1）)
3 * k (K + 1) (2k + 1) / 6 + 3 (AK + 1) & # 178; = (2k + 3) (1 + k) K / 2 + (2k + 3) * (AK + 1)
3 (AK + 1) & # 178; - (2k + 3) (AK + 1) - K (K + 1) = 0, that is ((AK + 1) - (K + 1)) * (3 (AK + 1) + k) = 0
So AK + 1 = K + 1
To sum up, an = n

How to prove permutation number formula by mathematical induction?
Thank you!

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