# Terminology - Nth rule

I'm trying to help my son study 8th grade math (in Texas). He's having trouble with the "Nth rule". Is anyone familiar with this terminology? I think that it may be related to the Nth term in a series, but I also think that it has a more specific meaning.

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As far as I can tell, "nth rule" means "a rule that gives the nth term in a series." (That is, it's a different rule for each series.) – Qiaochu Yuan Nov 28 '10 at 22:13

A quick google search suggests that "the $n$th rule" means finding the formula for the $n$th term of a given sequence of numbers.

For example, given $1$, $3$, $5$, $7,\ldots$, the expected "nth rule" is probably "$2n-1$", since the 1st term is $2(1)-1$, the 2nd term is $2(2)-1$, the 3rd term is $2(3)-1$, etc.

What follows is a bit of a diatribe completely irrelevant to your question: It is an utterly silly request, as noted long ago by Carl Linderholm in his book, Mathematics made difficult. Any answer that gives the right values at the given points is valid; and if you pick your favorite number for the "next" number, you can find a formula that will give those using Lagrange interpolation; for example, if you have the sequence $1$, $2$, $4$, $8$, $16,\ldots$, then the "next term" is $31$, either through Lagrange interpolation, or by counting the number of sectors that a circle is divided into if you place $n$ points on its circumference and connect them with straight lines. Both show much deeper mathematical understanding than "$2^{n-1}$". The problem is not really "What is the formula for the $n$th term of this sequence?" the question is "of all possible formulas that would give these values for $n=1,2,\ldots$, which is the one that the person who is grading this test is thinking of?"

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Totally agree. The space of functions to consider must be reduced, such as: Find the polynomial of lowest degree that interpolates all given values. – Lagerbaer Nov 28 '10 at 22:17

What they mean is this: Suppose I give you a sequence like

2,4,6,8,...

Then I ask you: How can I obtain the n-th term in this sequence?

And you'll say: It's just $2n$.

Now, for sequences of the form $an + b$, it is quite easy to obtain the rule: Just look how the element increases in each step, this gives you $a$. Then see what the first element is and choose $b$ so that it's obtained for $n = 1$.

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