Find the recursive definition for these two sets of numbers

Question

Good afternoon,

I'm struggling with finding the proper recursive definitions for these two sets of numbers:

problem 1: 1, –1, 2, –2, 4, –4, 8, –8, 16, –16, …

problem 2: 1, 2, 3, 6, 11, 20, 37, 68, 125, 230, 423,

I'm generally a dunce when it comes to problems such as these, especially when it involve positive/negative swaps like in the first set of integers, so I'd appreciate any help in solving these

Answer 1

For the first sequence, the terms of even index (assuming we start at the first term) are the negative of the previous, and the odd-indexed terms are $(-2)$ times the previous.

For the second sequence, the fourth term onward is determined by the sum of the previous three terms (also known as the "Tribonacci numbers", as noted by J.W. Tanner in the comments).

With these in mind, you should be able to express the relationships algebraically with ease.