  We derive an identity connecting any two second-order linear recurrence
sequences having the same recurrence relation but whose initial terms may be
different. Binomial and ordinary summation identities arising from the identity
are developed. Illustrative examples are drawn from Fibonacci, Fibonacci-Lucas,
Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas sequences and their
generalizations. Our new results subsume previously known identities.
