Question

Show your rationale and steps for each problem. Try to show as many steps/explanation as possible for full credit. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

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)

Answer 1

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