The aim of this paper is to obtain Binet formula for k-Jacobsthal numbers. And also with the help of Binet formula we obtain some properties for the k-Jacobsthal numbers.Mathematics Subject Classification: 11B37, 05A15, 11B83

The Fibonacci number is famous for possessing wonderful and amazing properties. In this study, we introduce the k-Fibonacci-Like number and related identities. We establish some of the interesting properties of k-Fibonacci-Like number. We shall use the Induction method and Binet's formula for derivation.

In this article, we introduce a new generalization of Fibonacci sequence and we call it as k-Fibonacci-Like sequence. After that we obtain some fundamental properties for k-Fibonacci-Like sequence and also we present some relations among k-Fibonacci-Like sequence, k-Fibonacci sequence and k-Lucas sequence by some algebraic methods.

In this paper, we derive various formulae for the sum of k-Jacobsthal numbers with indexes in an arithmetic sequence, say an r + for fixed integers a and . r Also, we describe generating function and the alternated sum formula for k-Jacobsthal numbers with indexes in an arithmetic sequence.

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