Categories of Sequences

Today, we are going to learn about strictly increasing sequences. Can anyone tell me what a strictly increasing sequence is?

Is it a sequence where each number is larger than the previous one?

Exactly! So, if we have a sequence that starts with 1 and ends with n, like 1, 2, ..., n, how might we represent that with a function?

Maybe we can use a function that counts these sequences?

That's right. We will denote this function as a(n), representing the number of valid sequences that end with n.

How do we calculate that?

Great question! We will derive a recurrence relation to do that. But first, let’s categorize these sequences based on their second last term.

Remember the acronym SEQUENCE to help you remember: Start, End, Quarters, Unique, Numbers, Construction, Endings.

That’s helpful!

Let’s explore how we derive this recurrence. First, we consider Category 1, where the second last term is n-1. How many sequences can we have there?

I think we can take all sequences that end with n-1 and just add n at the end?

Exactly! So that gives us a(n-1) sequences. Now, what about Category 2?

In Category 2, the second last term can be any number from 1 to n-2, right? So we'd count those sequences too.

Perfect! And if we sum these two categories, we can express it mathematically as a(n) = a(n-1) + ... + a(2). Let’s consolidate this into a more compact form.

So we can simplify it to just rely on the last calculated term?

That's right! It shows that a(n) depends only on a(n-1), leading us to a linear recurrence relation. Excellent insight!

Use the mnemonic 'COMPASS' for 'Compact, Organized, Mathematical, Practical, Aspect, Sequence, Structure' to remember how we approach recurrence equations.

As we found our recurrence, we need to discuss initial conditions. Why do you think they’re important?

Without them, we can't start our calculations or will have inaccurate answers.

Exactly! For instance, what would a(1) equal?

That should be 1 since there's only one sequence with just a single number.

Great! And what about a(2)?

Also just 1, right? The only sequence being {1, 2}.

Spot on! We need these base cases to function correctly when applying our recurrence relation. It’s such a crucial step!