To understand how recursive formulas work for linear and quadratic sequences, let's break it down into simpler parts.

1. Linear Sequences:

These sequences grow by adding the same number each time.

For example, take the sequence: 2, 4, 6, 8.

We can write the recursive formula like this:

• First term: (a_1 = 2) (this is the starting number)
• Next terms: (a_n = a_{n-1} + 2) for (n > 1) (this means each new term is 2 more than the one before)

So, to get from 2 to 4, you add 2. To get from 4 to 6, you add 2 again.

2. Quadratic Sequences:

In these sequences, the growth changes but follows a specific pattern.

For example, look at this sequence: 1, 4, 9, 16.

Here’s how we can write the recursive formula:

• First term: (a_1 = 1)
• Next terms: (a_n = a_{n-1} + 2n - 1) for (n > 1) (here, the amount you add grows larger by 2 each time)

So, to get from 1 to 4, you add 3. To get from 4 to 9, you add 5. The amount you add keeps getting bigger!

By using these formulas, you can easily create sequences step by step!