An explicit rule for the sequence is: [a_n = n^2 - n]
To write an explicit rule for the given sequence 1, -2, -3, -8, we need to identify the pattern or relationship between the terms. In this case, the pattern is not immediately obvious, so we can use the differences between the terms to help us find the rule.
The first differences between the terms are -3, -1, -5. The second differences are 2, -4. Since the second differences are not constant, the sequence is not quadratic.
One possible rule for the sequence is that each term is the result of subtracting the position of the term from the square of the position of the term. For example, the first term is 1, which is 1 - 1, the second term is -2, which is 4 - 2, the third term is -3, which is 9 - 3, and the fourth term is -8, which is 16 - 4.
Therefore, an explicit rule for the sequence is: [a_n = n^2 - n]
where (a_n) is the (n)th term of the sequence.