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1.) Find an explicit non recursive formula for the nth term (an) of the sequence. 1) 0, -1, 0, 1, 0, -1, 0, 1 (0, -1, 0, 1 repeated)
The sequence 0, -1, 0, 1 repeats in a pattern. We can observe that the index n follows a pattern: 1, 2, 3, 4, 5, 6, 7, 8.
If we divide n by 3 and take the remainder, we get the terms of the sequence: 0 divided by 3 has a remainder of 0, 1 divided by 3 has a remainder of 1, 2 divided by 3 has a remainder of 2, 3 divided by 3 has a remainder of 0, and so on.
To find an explicit non-recursive formula for the nth term (an) of the sequence, we can use the modulo operation.
If we denote the remainder of n divided by 3 as r, then the nth term (an) can be expressed as follows:
an = r - 1
For example:
- For n = 1, the remainder of 1 divided by 3 is 1, so a1 = 1 - 1 = 0.
- For n = 6, the remainder of 6 divided by 3 is 0, so a6 = 0 - 1 = -1.
Therefore, the explicit non-recursive formula for the nth term (an) of the sequence 0, -1, 0, 1, 0, -1, 0, 1 (0, -1, 0, 1 repeated) is:
an = (n mod 3) - 1