Recursive Rules

Recursive. That word makes us think of second grade when teachers forced us to write with the cursive m that had three humps instead of two. Whoever came up with that? Either way, recursive rules have nothing to do with cursive handwriting. In fact, "recursive" is an adjective that describes a certain type of rule. It comes from the word "recursion," which is fancy lingo for something that's generated from prior or preceding terms.

In plain English, a recursive rule is when we find the next term of a sequence using the one (or ones) before it. No. Big. Deal. It can actually be easier sometimes.

To figure this out, we're going to go back to the sequences we already created rules for. This time, we'll make recursive rules instead of explicit ones.

Sample Problem

Find the recursive rule and a₆ for the sequence: 4, 7, 10, 13, …

This time, we want to find a rule that uses a_{n − 1} to get a_n. This is nothing more than the mathematically cool way of saying we want to use the second term to get the third, or the fifth term to get the sixth.

With this sequence, we take each term and add 3 to it. So we'll take the term that we want to know, a_n , and write it in terms of a_{n − 1}, or the term that came right before it.

{a_n} = (a_{n – 1}) + 3

The biggest downfall of recursive rules is that in order to find another term, like a₆ in this case, we need to first find a₅ and a₄ and so on down the line. There's no plugging in n like we did before.

In this example, a₄ = 13, so a₅ = 13 + 3 = 16 and a₆ = 16 + 3 = 19. So a₆ = 19.

Not too bad. But what if we were asked for a₃₀? Holy terms, Batman.

Sample Problem

Find the recursive rule and a₁₀ for the sequence 2, 4, 8, 16, …

Each term is multiplied by 2 to get the next term. This sets up B-E-A-utifully. Our rule couldn't be much finer:

{a_n} = 2(a_{n − 1})

The problem we do run into is finding a₁₀ for this guy. We only have four terms, so we'll have to do some serious multiplication. Here it goes…

a₅ = 2(a₄ ) = 2(16) = 32
a₆ = 2(a₅ ) = 2(32) = 64
a₇ = 2(a₆ ) = 2(64) = 128
a₈ = 2(a₇ ) = 2(128) = 256
a₉ = 2(a₈ ) = 2(256) = 512
a₁₀ = 2(a₉ ) = 2(512) = 1024

So a₁₀ = 1024, just like it did the first time.

Sample Problem

Find the recursive rule for the sequence 0, 1, 1, 2, 3, 5, 8, 13, …

This sequence is very famous and is known as the Fibonacci sequence. Without going into too much detail, this sequence is found all over the place, literally, and is really awesome. But don't take our word for it; check out this video. (You're going to want to watch this one.)

Now let's answer the question. We wanna find a recursive rule. This time, there is no common number added, subtracted, multiplied, or divided. There are no exponents involved. And, try as we might, there are no factorials. Instead, starting with the third term, each term in the Fibonacci Sequence is based solely on the previous two terms.

a₁ = 0
a₂ = 1
a₃ = a₂ + a₁ = 1 + 0 = 1
a₄ = a₃ + a₂ = 1 + 1 = 2
a₅ = a₄ + a₃ = 2 + 1 = 3

And so on.

The rule must take into account both a_{n − 1} and a_{n − 2} which gives us:

{a_n} = (a_{n – 1}) + (a_{n – 2})

Congratulations. You have officially graduated into the realm of the mathematically informed. More importantly, you now understand why this joke is funny:

A math professor walks into a Fibonacci conference, looks at an esteemed math professor colleague and says, "Man, I can't wait. I think this year's conference is going to be as big as the last two years combined."

Bonus Problem

Fibonacci's Honeybees

In a beehive there are male and female bees. There's a catch, though. Female bees are born when a male bee fertilizes a female egg. Male bees are born when a female egg is left unfertilized. This means males have a mother but no father. Weird. Your mission: draw the first six levels of the family tree of a male honeybee.

The first level has just one male, the one whose tree we are drawing. The second level has just one bee, the male's mom. The third level has a male and female since each female has a mom and a dad. The next level will have a male and female pair in addition to another female—and so on and so fourth.

Check it out. The levels of the honeybee family tree are 1, 1, 2, 3, 5, 8, … #iKnowRight