What is the range of a sequence?

Unlocking the Secrets of Sequence Ranges: It’s Simpler Than You Think!

Sequences. They might sound intimidating, like something only mathematicians care about. But trust me, they’re actually pretty fundamental, popping up all over the place from simple counting to super complex calculations. And one cool thing about them is their “range.” So, what exactly is the range of a sequence? Let’s break it down in plain English.

First things first, what’s a sequence anyway? Think of it as an ordered list – usually of numbers, but it could be other things too. It’s like a line of dominoes, each one following the next in a specific order. We can write them out like this: a₁, a₂, a₃… where each “a” with a little number is a term in the sequence.

Now, sequences can either end (finite) or go on forever (infinite). A simple example of a finite sequence is: 2, 4, 6, 8, 10. An infinite one? How about: 1, 1/2, 1/3, 1/4, and so on, getting closer and closer to zero.

Okay, now we can talk about the range. Ready? The range of a sequence is simply the set of all the different values you see in the sequence. That’s it! Think of it as collecting all the unique dominoes that actually fell, ignoring any duplicates. The range is a set, which is math-speak for a collection where the order doesn’t matter, and you only list each item once.

Let’s look at some examples to make it crystal clear:

It’s super important to remember that the sequence is different from its range. The sequence is the entire ordered list, while the range is just the unique values that appear in that list. The sequences (1, 1, 2, 2, 3, 3) and (1, 2, 3, 1, 2, 3) are different, but they both share the same range: {1, 2, 3}.

So, why should you care about the range? Well, it tells you a lot about the sequence’s behavior. For example, it helps us figure out if a sequence is “bounded” or not. A bounded sequence is one where all the terms are stuck between two numbers – they don’t go off to infinity. Think of it like a dog on a leash; it can move around, but it can’t run away completely.

• The sequence {1/n} is bounded because all the terms are between 0 and 1.
• But the sequence {n} is unbounded – it just keeps growing forever!

And that brings us to why the range is actually useful!

Think about it:

In short, the range of a sequence is a simple but powerful idea. It’s the set of all the unique values in the sequence, and it helps us understand how that sequence behaves. So, next time you see a sequence, don’t be intimidated – just think about its range! It’s like a secret decoder ring for understanding the sequence’s story.