Among several famous and interesting sequences that exist in mathematics, Fibonacci sequence plays a prominent and significant role. It possesses wide applications that were used or observed in several areas of science and technology. Nature exhibits several objects based on Fibonacci sequence. The general formula for determining Fibonacci numbers is usually obtained through the recurrence relation. In this paper, we try to extend the recurrence relation of Fibonacci sequence to cover more general class of sequences in two different ways. The ratio of (n + 1)th term to the nth term, which is called the limiting ratio of each such sequence is computed and the limiting ratios for different values of the generalized parameter is discussed in detail. The results arrived in this paper will be exciting as well as rewarding

Fibonacci sequence, Generalized Recurrence Relation, First Fibonacci type sequence, Second Fibonacci type sequence, Limiting Ratio.