First order linear recursive sequences

A first order linear recursive sequence is a sequence of numbers that is generated by

1. choosing a starting number,
2. choosing two numbers k and l, such that k ≠ 0,
3. constructing the next number from the previous number by multiplying the previous number by k and adding l.

Or if you are a bit more mathematically inclined

A first order linear recursive sequence is a sequence of numbers a₁, a₂, a₃, …. such that there exist numbers k and l such that k ≠ 0 and

a_n+1 = k a_n+ l for n ≥ 1.


Or, if you are the author of "Math Makes Sense 6":

There is no definition of what a pattern rule is. How do I know that the last definition defines first order linear recursive sequences? I am not 100% sure, but there is some evidence given by the trial and error method.  Firstly, all the sequences that were recursive by the above definition were first order linear recursive sequences. Secondly, there were some recursive sequences that there were declared not recursive by the above definition. For example

1, 4, 9, 16, 25, 36,...   is declared not recursive, whereas it is first order recursive because
a_n+1 = (sqrt (a_n)+1)².

1, 7, 14, 20, 27, 33, 40, .... is not recursive, even though is second order liner recursive because 
a_n+1 =  a_n-1+ 13.

(The above two examples were given for homework and marked as stated by the teacher, who I presume has the solution manual provided by the authors of the book).

The teacher said that the second example is an example of an alternating pattern. I don't understand  how a child can understand why the following rule is not a good rule to make a pattern recursive:

Start at 1, alternately add 6 and 7.

My favourite homework problem is problem x on page y.

x. How many recursive patterns are there whose first two terms are 4 and 7? Write the first 5 terms of the ones you find.

My daughter asked me to check her work. What came to my mind when I read the question was: "How long is this homework supposed to take?" because the answer to the first questions is:
'There are infinitely many recursive patterns that start with 4, 7. When I checked what my daughter wrote, there were 3 examples. Then the conversation went something like this:

Me: You didn't answer the first question here. How many recursive patterns that start with 4 and 7 are there?
She: Oh, I don't need to answer this question. I gave 3 examples.
Me: Well, how many examples can you give?
She: As many as they want.
Me: Yes, that's right. Why didn't you say that?
She: I don't need to say that, they just want 3 examples.
Me: Usually, when there are many parts of a question, you need to answer them all.
She: You just don't know how this works. I have been doing this for 3 years. I always had an A in Math. You don't know how to answer their questions.
Me: a-aaa--aa

What could I say? She was right. But then, I remembered something else. When I give my students (post secondary) a question that consists of several parts, I need to split it into parts a), b), c)... Otherwise, quite  lot of students don't answer all the parts. Also, sometimes when I ask them to prove something, they just give a few examples. Now, I know where the root of the problem is.

I don't understand why the authors of "Math Makes Sense" don't use the proper terminology. First order linear recursive sequence is long, but you can abbreviate it to FOLR sequence, if you need to. But remember that your audience are the kids who have no issues with words like Tyrannosaurus rex and Pterodactyl.

When you introduce FOLR sequences, it is likely OK to ask the students to eyeball which ones are FOLR and which ones are not. This probably increases students understanding. But Math is about figuring out how to systematically solve problems, and there is not much systematic about eyeballing. I don't understand why a method for identifying which sequences are FOLR sequences, and which are not, is not presented. Namely, if you write the following two equations:

a₃ = k a₂+ l
a₂ = k a₁+ l

and subtract the second one from the first one, you get

a₃ - a₂= k (a₂- a₁)

from which you can calculate k as k = (a₃ - a₂) / (a₂ - a₁).

After that you can calculate l from the equation a₂ = k a₁+ l and see if the recursion works for the rest of the given sequence.

I understand that the above algebraic manipulations are beyond the grade 6 students, but this whole lesson is on patterns, and so you can ask the students to take 3 recursive sequences and construct from them new sequences by calculating the values  b_n = a_n+1 - a_n for n ≥ 1. Then construct the third sequence c_n = b_n+1 / b_n

For example for a sequence

1, 3, 7,  15, 31, 63,.....

construct the sequence b:

2, 4, 8, 16, 32, ....

and the sequence c:

2, 2, 2, 2, .....

If the original sequence is FOLR, the the sequence c is always going to be constant and consist of the multiplier k.

If the students do the above procedure three times, they will easily figure out the pattern. They will also develop some algorithmic thinking. I see no possible setbacks. Do you?