# How to write a sequence formula

This right over here is a sub 4. So this is clearly an arithmetic sequence. For the second term, we added 4 once. We'd call it a sub 2. The answer lies in the concept of mathematical proof. But I could use the notation b sub k or anything else.

The method described here will not work for sequences like this one that are not polynomial sequences.

So just to be clear, this is one definition where we write it like this, or we could write a sub n, from n equals 1 to infinity. But I want to make us comfortable with how we can denote sequences and also how we can define them. Starting at k, the first term, going to infinity with-- our first term, a sub 1, is going to be 3, now.

Nobody, not even an experienced mathematician, writes a complete, well-written proof from start to finish the first time they try.

## Sequence calculator

In this case, d is 7. This is an explicit function. So how would we do that? So it's going to be 1 plus 3, which is 4. So it's going to be the previous term plus whatever your index is. Well, it's going to be a sub 2 plus 3. This right over here is a plus 3.

There are some conjectures in math that have eluded all attempts at proof for hundreds of years. So a given term is equal to the previous term. That's how much you're adding by each time. When k is 3, we get 7.

### Finding a formula for a sequence of numbers

So the same exact sequence, I could define it as a sub k from k equals 1 to 4, with-- instead of explicitly writing the numbers here, I could say a sub k is equal to some function of k. So for the second term, we add 2 once. We could either define it explicitly, or we could define it recursively. Now, how does this make sense? And then for n is 2 or greater, a sub n is going to be equal to what? That's how much you're adding by each time. This is another way of defining it.
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