Inverso-Fibonacci

     Many students know about the topic of Fibonacci sequences, where after two terms are given, each succeeding term is found by adding the previous two terms. Though the first two numbers could be anything, the main sequence begins with a pair of 1's.

1, 1, 2, 3, 5, 8, 13, 21,…

     This is expressed algebraically as a₁ = 1, a₂ = 1, a_n = a_n-1 + a_n-2.

     Of course, the first two terms may even be different values, thus leading to more interesting levels in the study of patterns.

     Little known, however, is what might be called an Inverse Fibonacci sequence, where the addition sign is replaced by subtraction. This simply means that once the first two numbers are given, each succeeding number is given by

a_n = a_n-1 - a_n-2.

     Given that the first two numbers of an I-F sequence are 4 and -7, what is the sum of the first 2004 terms of this sequence?

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     Extra: If you worked this problem carefully, you perhaps observed that there is a "trick" or pattern involved. Prove that this is indeed true, given that a₁ = x and a₂ = y.