  In this paper we present a family of identities for recursive sequences
arising from a second order recurrence relation, that gives instances of
Zeckendorf representation. We prove these results using a special case of an
universal property of the recursive sequences. In particular cases we also
establish a direct bijection. Besides, we prove further equalities that provide
a representation of the sum of $(r+1)$-st and $(r-1)$-st Fibonacci number as
the sum of powers of the golden ratio. Similarly, we show a class of natural
numbers represented as the sum of powers of the silver ratio.
