Final answer:
The Lucas sequence is not strictly monotonically increasing for all n ∈ N as the second term is less than the first. However, from the third term (n ≥ 3) onwards, it does become strictly increasing as each term is the sum of its two predecessors and greater than the term before it.
Step-by-step explanation:
The Lucas sequence is related to the Fibonacci sequence, where each term is the sum of the two preceding ones. The sequence generally begins with 2 and 1 (L0=2, L1=1), so the question refers to whether the Lucas sequence is strictly monotonically increasing for all n ∈ N, or just for n ≥ 2. By observing the first few terms (2, 1, 3, 4, 7, ...), we can see that the sequence is not strictly increasing from the start since the second term is smaller than the first. However, from the third term onward, the sequence does become strictly monotonically increasing, as each term is larger than the one before it.
To demonstrate this, consider any term Ln for n ≥ 3 in the Lucas series; it is the sum of the two previous terms Ln = Ln-1 + Ln-2, where Ln-1 and Ln-2 are positive. Since Ln-1 > Ln-2 for n ≥ 3, it follows that Ln > Ln-1, confirms the strictly monotonically increasing nature of the sequence from the third term onwards.