After multiplying the first and last terms in the 3 numbers of the fibonacci sequence and subtracting their product by the square of the middle number I get 1 as the difference, but sometimes the subtraction is the opposite, is there a special sequence ?

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## Answers (2)

If you are taking the sequence as 1, 1, 2, 3, 5, 8, 13, ......... with the first term 1, the second term 1, the third term 2, and so on, it appears that if the terms you multiply are odd numbered terms (subtracting the square of an even term) the result is +1.

If the terms you are multiplying are even numbered terms (subtracting the square of an odd term) the result is - 1.

This is based on inductive reasoning only, and in no way constitutes a proof.

So fibonacci is: 1 1 2 3 5 8 13 21 34 55 89

The next number is found by taking the sum of the previous two numbers.

So let's try a few:

1 * 2 - 1^2 = 1

1 * 3 -2^2 = -1

2 * 5 - 3^2 = 1

3 * 8 - 5^2 = -1

5 * 13 - 8^2 = 1

8 * 21 - 13^2 = -1

so what are we seeing here? an alternating sequence of positive 1, negative 1, positive 1, negative 1 repeating. so that is your special sequence. maybe your teacher wants you so write it as (-1)^(n+1)