### On Generalized Fibonacci like Sequence Containing Higher Powers

Krishna Kumar Sharma

#### Abstract

In mathematical terms, sequence is a list of numbers arranged in a specific order. Fibonacci sequence  is sequence of whole numbers defined by the recurrence formula Fn=Fn-1 + Fn+2, n>=2 and F0=0, F1=1 where Fn is an nth number of sequence. Fibonacci numbers and their generalization have many interesting and surprising properties. Many authors have defined Fibonacci pattern-based sequences which are popularized and known as Fibonacci-Like sequences. When the first two terms of the Fibonacci sequence become arbitrary, it is known as Fibonacci-like sequence. Fibonacci-like sequence can start at any desired number. This study aimed to derive and validate a formula in solving Fibonacci-like sequence. Here we introduced a generalized Fibonacci Like sequence defined by Gn+1 + bGn-1, n>=1 and G0=0, G1=s where a, b and s are real numbers. In this research paper we introduced and derived various analytical properties. These identities of Generalized Fibonacci like sequence containing higher powers have been proved by Binet’s formula and other techniques.

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