# Tweaking the Fibonacci Sequence!

Calculus Level 5

Let $$k \in (0,1)$$, and let the sequence $$(B_n)_{n=0}^{\infty}$$ be defined by:

$$B_0 = k, B_1 = k^2$$ and $$B_{n+2} = kB_{n+1} + k^2B_n$$ for integers $$n \geq 0$$.

If the value of the expression $$\large{\displaystyle \sum_{n=0}^{\infty} \dfrac{B_n}{n+1}}$$ can be expressed as:

$\large{\dfrac1{\sqrt{\gamma}} \ln \left(\dfrac{1-\beta k}{1-\alpha k}\right)}$

where $$\gamma \in \mathbb Z; \alpha, \beta \in \mathbb R$$ and $$\alpha >0, \beta<0$$. Find the value of $$\large{\lfloor \alpha + \beta + \gamma \rfloor}$$.

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