MathDB
f_n = (a^{n+1} - b^{n+1})/\sqrt5, where a, b are real such a+b=1, ab=-1, a>b

Source: Vietnamese MO (VMO) 1964

August 22, 2018
recurrence relationalgebraInteger sequence

Problem Statement

Define the sequence of positive integers fnf_n by f0=1,f1=1,fn+2=fn+1+fnf_0 = 1, f_1 = 1, f_{n+2} = f_{n+1} + f_n. Show that fn=(an+1−bn+1)5f_n =\frac{ (a^{n+1} - b^{n+1})}{\sqrt5}, where a,ba, b are real numbers such that a+b=1,ab=−1a + b = 1, ab = -1 and a>ba > b.
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