May 30, 2017

Quadratic Sequences: How to Find Explicit and Recursive Definitions

Your math book probably doesn't explain how to get explicit and recursive definitions of quadratic sequences. Most of the solutions on the Internet involve systems of three equations. Fortunately, I've come up with something simpler.

Quadratic Sequences

A sequence is quadratic if the second difference, also known as the difference of the difference, is constant. In the picture below, the second difference is equal to 2, and it's constant, so the sequence is quadratic.
Note that the first difference is just the slope of whatever quadratic function the sequence comes from. If the first difference is the slope, that means the second difference is the slope of the slope.

Getting Explicit Definitions

To get an explicit definition, we need to make the sequence above fit a quadratic function:


At this point, you've probably been told to create a system of three equations using f(1) = 5, f(2) = 10, and f(3) = 17 in order to solve for a, b, and c.

I'm happy to tell you that there's an easier way.

Astute Calculus AB students may already have noticed that the second slope is just the second derivative of f(n). In other words,

The second difference is equal to 2a.

(If you're not in Calc AB, write down the bold-faced words in the previous sentence.)

That means we can figure out that a = 1 just by looking at the sequence!

We an also figure out c just by looking. If we plug in n = 0, we get f(0) = c. In other words,

The constant c is equal to the n = 0 term of the sequence.

We can work backward from f(1) to get to f(0). Kudos to one of my students for pointing this out!

Again, we've figured out that c = f(0) = 2 just by looking at the sequence!

Since we know that a = 1 and c = 2, our closed definition is almost complete:


Now we can get b by plugging in one of the terms from the sequence.

Let's plug in f(1) = 5:


After we simplify and solve, we'll get b = 2 along with the complete explicit form of the sequence:


You can check this definition by regenerating the original sequence starting at n = 1. It works!

Here's a quick summary of what you need to know to get the explicit form of a quadratic sequence:

  1. The second difference is equal to 2a.
  2. The constant c is equal to the n = 0 term of the sequence.
  3. Get by plugging in one of the terms from the sequence. 

Getting Recursive Definitions

Here's the sequence again in case you need it:


The recursive definition of a quadratic sequence has the form


The first part of the definition is the first term of the sequence: f(1) = 5. It's easy, but don't forget to write it down when you do problems on your test!

The second part is almost as easy. Remember that the second difference is equal to 2a, so just put the second difference in front of n. (If you're starting with the explicit form, you can just multiply a by 2 and stick the resulting number in front of n.)


Now we can just plug in terms from the sequence and solve for the constant d:
Knowing d gives us the complete explicit definition:

f(1) = 5
f(n) = f(n-1) + 2n + 1

You can check this definition by regenerating the original sequence starting at n = 2. It works!

You can also check the recursive definition intuitively. Remember that the first difference is the slope of the sequence. Our recursive definition says that the slope should be 2n + 1, which matches the first difference we got from inspection. The second difference should be the slope of the slope. The slope of the slope is just the slope of 2n + 1, which is 2, which also matches the second difference we got by inspection. If you're in Calculus AB, you'll note that we just took a second derivative.


Here's a quick summary of what you need to know to get the recursive form of a quadratic sequence:

  1. The first part of the definition is the first term of the sequence.
  2. Put the second difference in front of n.
  3. Plug in terms from the sequence and solve for the constant d.
I wish you hours of fun as you study for finals!

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