What is a general term and how is it different than a function?

[priceofcarrot]

Homework Statement

Homework Equations

The Attempt at a Solution

I don't know how to use the above template considering I don't actually have a specific question from my homework, but I do have a question that I need answered to complete my homework.

Anyway, my book asks me to write a function that would correspond to a sequence, and then later asks me to write a general term for the nth term of that same sequence.

Would these be the same answer? I wrote them down as the same answer just with n instead of x for the general term question. They also asked me for a recursive formula for the sequence.Can a general term be a recursive formula or a non recursive formula?

Thanks in advance. I just really don't understand what a general term is. Is it possible that they want the same answer for the general term and recursive formula, or function?
My problem is really that after writing a function, and a recursive formula for a sequence, I can't think of any other way to write a formula differently that could be considered a general term. I don't know what a general term is in the first place though.

 

[HallsofIvy]

So your question is about the "general term of a sequence". If a sequence is, for example, 1, 1/2, 1/3, 1/4, 1/5, ... so each term is of the 1/n for some n. The "general term" is "1/n".

No, a "term" is not a "formula" but a formula might involve one or more terms. For example, a recursion formula typically says that a given term is some function of the previous terms. For example (although its a bit of a stretch), we can write $a_{n+1}= 1/(n+1)$ as $(1/n)- 1/(n(n+1))$ so we can write $a_{n+1}= a_n- 1/(n(n+1))$, a recurrance formula.

My problem is really that after writing a function, and a recursive formula for a sequence, I can't think of any other way to write a formula differently that could be considered a general term.

Perhaps it would help if you gave a specific example. What "function" are you talking about?

 

[priceofcarrot]

Okay, so this specific question has
x = 1, 2, 3, 4
f(x) = 0, 3, 8, 15

for this sequence's function I wrote x^(2) - 1.

Is the general term the same answer but with an n instead of x?

thanks

 

[dimension10]

Okay, so this specific question has
x = 1, 2, 3, 4
f(x) = 0, 3, 8, 15

for this sequence's function I wrote x^(2) - 1.

Is the general term the same answer but with an n instead of x?

thanks

Yup, you don't really need to change the $x$ to an $n$, actually. It is just a convention. It is like:

$$a_n=n^2-1$$

 

[priceofcarrot]

Thanks a lot. It's very helpful to get an answer to this.