In this paper we investigate several properties and characteristics of the generalized Fibonacci sequence

={a, b, a+b, a+2b, 2a+3b, 3a+5b,...}. This concept is a generalization of the famous Fibonacci sequence. In particular we find the identities of sums and the nth term

in detail. Also we find the generalizations of the Catalan's identity and A. Tagiuri's identity about the Fibonacci sequence, and investigate the relation between

and Pascal's triangle, and how fast

increases. Furthermore, we show that

and

are relatively prime if a b are relatively prime, and that the sequence

of the ratios of consecutive terms converges to the golden ratio

.