Question

Consider the sequence given by a₁ = 1, a2 = 1, and

aₙ = aₙ₋ₐₙ₋₁ + aₙ₋ₐₙ₋₂
Determine a₁₆
(Note: In previous assignments I encouraged finding a pattern and extrapolating. That won't work here.)

Answer 1

To determine a16 in the given recursive sequence, each term must be calculated manually from a1 and a2 given that each term depends on the values of the two previous terms. There is no discernible pattern or shortcut; thus, a step-by-step approach is necessary.

Step-by-step explanation:

The sequence described in the student's question is defined recursively, with each term depending on the preceding two terms in a unique manner. To find a16, one must calculate each term sequentially from the beginning, since each term relies on the positions determined by the values of the two previous terms.

To calculate a16, we begin by noting a1 = 1 and a2 = 1. For a3, since a2 = 1 and a1 = 1, we have a3 = a3-1 + a3-2 = a2 + a1 = 1 + 1 = 2. The process must be continued by calculating each subsequent term based on the recursive formula.

This method of defining a sequence does not allow for a simple pattern recognition or a closed-form solution, and therefore each individual term must be calculated manually or through a computational algorithm designed to handle such recursive sequences.