New Results on k-Fibonacci Numbers and Powers of its Golden Ratio Expressed as Continued Fraction

This paper uses the tools of two very important branches of Number Theory Continued Fractions and theory of Fibonacci numbers. The Fibonacci sequence has been generalized in many ways, some by preserving the initial conditions, others by preserving the recurrence relation. The relationship between the golden ratio and continued fractions is commonly known about throughout the mathematical world. The convergents of the continued fraction are the ratios of consecutive Fibonacci numbers. The continued fractions for the powers of the golden ratio also exhibit an interesting relationship with the Fibonacci numbers. The ratios of any k-Fibonacci sequence {, } is expressed by means of continued fraction. We find simple closed form continued fraction expansions for, for any positive integer r.