Generalization is my frenemy

Number Theory Level 4

Consider the generalization of the Fibonacci sequence: define the $n$ -bonacci sequence such that its term is sum of the previous $n$ terms with initial terms: $a_{0,n} = a_{1,n} = a_{2,n} = \ldots = a_{n-2,n} = 0, a_{n-1,n} = 1$ .

Now we denote the $m^\text{th}$ term of the $n$ -bonacci sequence as $W_{m,n}$ . Calculate the limit below.

$\large \lim_{n \to \infty} \lim_{m \to \infty} \frac { W_{m+1,n} }{W_{m,n}}$

Details and Assumptions :

As an explicit example: if $n=2$ , we will get a Fibonacci sequence, which means the limit is $\phi = \frac { 1 + \sqrt 5}{2}$ .
You do not need computational aids to solve this problem.

Inspirations: Problem 1 and Problem 2 .

The answer is 2.

1 solution

Samuel Li
Apr 5, 2015

Let $\lim_{m \to \infty} \frac { W_{m+1,n} }{W_{m,n}} = R$ . It is easily shown that $R^n = R^{n-1} + R^{n-2} + R^{n-3} + \ldots + 1$ . Using the formula for a geometric series, this simplifies to $R^n = \frac{R^n-1}{R-1}$ . Rearranging, we get $R^{n+1} -2R^n+1=0$ . Dividing by $R^n$ , $R - 2 + \frac{1}{R^n} = 0$ . Taking the limit as $n \to \infty$ , $R-2 = 0$ , or $R=\boxed{2}$ .

Generalization is my frenemy

Inspirations: Problem 1 and Problem 2 .

The answer is 2.

1 solution

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