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If $T_n = n(n+2)$ for a given sequence, verify that the sequence is not arithmetic.

I can't figure out how to substitute $T_{n+1}$ for $n$ correctly and work out the brackets afterwards in a good order.

StubbornAtom
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Alpha
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    As I suggested in your prior question, please write out the first few elements in your sequence. That should, at least, give you something to check your work against. Also, to show that the sequence is not arithmetic, all you need to do is to show that the differences between successive terms are not constant. – lulu Aug 28 '16 at 23:07
  • I understand that, but it doesn't unfortunately show me how to use the formula afterwards :/ knowing the elements in the next sequence which isn't obviously hard at all doesn't help me in understanding the formula as the difference contains n term anyway – Alpha Aug 28 '16 at 23:11
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    Please edit your post to reflect that you can at least solve the problem that way. As it stands, there is no evidence that you have tried anything at all. – lulu Aug 28 '16 at 23:12
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    The difference is constant if and only if the sequence is arithmetic by definition. Is the difference constant. Do you know what $T_{n+1}$ is? What about $T_{n+1}-T_{n}? Is it constant? – Ahmed S. Attaalla Aug 28 '16 at 23:14
  • I know that T1 is 3, T2 is 8, T3 is 15, T4 is 24 but knowing that I still have no idea how to calculate it with the formula Tn+1. I tried substituting it for both the n's and then proceeding with multiplying by brackets but I just get confused, I'm really not looking for just the answer which I think you're implying that I'm trying to do just that, I want to realise how to do these questions – Alpha Aug 28 '16 at 23:18
  • Ok, step by step. What is $T_{n+1}$ in terms of $n$? Keep in mind that $T_z=z(z+1)$ no matter what $z$ is (well, supposing $z$ to be a natural number). – lulu Aug 28 '16 at 23:19
  • if Tn = n(n+2) then I guess Tn+1 would be n+1(n+1+2)? or (n+1)(n+1+2) – Alpha Aug 28 '16 at 23:22
  • @Alpha That's right (the second one - you need the parentheses!). Now what is $T_{n+1}-T_n$? (HINT: expand it out . . .) – Noah Schweber Aug 28 '16 at 23:23
  • Good, for the second one. We have $T_{n+1}=(n+1)(n+3)$. Can you write out the product? – lulu Aug 28 '16 at 23:24
  • that would be n^2+4n+4? for the two brackets I mean – Alpha Aug 28 '16 at 23:26
  • @Alpha Almost - check the constant term . . . (what is $3$ times $1$?) – Noah Schweber Aug 28 '16 at 23:28
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    @Alpha you seem averse to checking numerical values. I can't understand that. Checking values is critical in working algebraic problems. – lulu Aug 28 '16 at 23:30
  • oh yes, sorry, it's supposed to be n^2+4n+3, which would mean that n^2+4n+3 - n^2+2n = 2n+3 – Alpha Aug 28 '16 at 23:32
  • I'm very sorry for my lack of knowledge guys, I've missed a lot of my last year in school and now I'm trying to catch up. Thank you for helping – Alpha Aug 28 '16 at 23:33
  • @Alpha That's it exactly. (I'm curious - does the edit to my answer make it more helpful? I've found that a number of my students have difficulty plugging in expressions, and I'm currently not really sure how best to help.) – Noah Schweber Aug 28 '16 at 23:33
  • Take greater care in what you write. Your last comment is wrong as you have written it. You meant to write $n^2+4n+3-(n^2+2n)$. Leaving out the parenthesis matters! It changes the sign of the last term. – lulu Aug 28 '16 at 23:35
  • okay, sorry lulu, you're right. Noah, if you mean the comment with 3 times 1 then I only noticed it after I figured it's supposed to be 3, but on the other hand in my opinion it's good that you wrote that, maybe you should wait a bit more with your students and try to give them such bigger hints after more minutes than you did here but overall it's good that you know that you need to give a little bit more help and not put the student in an awkward situation – Alpha Aug 28 '16 at 23:38

1 Answers1

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In order to show that a sequence is not arithmetic, you just need to show that the differences between successive terms are not always the same; that is, you just need to find two terms $T_a$ and $T_b$ such that $$T_{a+1}-T_a\not=T_{b+1}-T_b.$$ That is, you're looking for a counterexample.

So let's compute some terms:

  • $T_1=1(1+2)=3$.

  • $T_2=2(2+2)=8$.

  • $T_3=3(3+2)=15$.

Do you see how to show that the sequence is not arithmetic, now?

Based on your comments, I suspect you are misunderstanding what "$T_n=$[stuff]" means; it means, in order to find $T_n$, plug $n$ into [stuff].

I think this is confusing because the same symbol "$n$" is getting used a lot. Think about it this way:

  • Think of $T_x=x(x+2)$ as a rule: take in a number $x$, and output the number $x(x+2)$.

  • So for instance, you've already calculated a few specific values of $T_x$.

  • But we can also calculate examples of $T_x$ where we plug a complicated expression, involving variables,in for $x$! For example, $T_n=n(n+2)$ . . .

  • And $T_{2y}=2y(2y+2)$ . . .

  • And $T_{c^3+30c-2}=(c^3+30c-2)(c^3+30c-2+2)$ . . .

  • and so on.

Does this help?

Noah Schweber
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  • yes, i see that, and thank you very much, and I really do understand that, I just don't get how the solution in my book says the difference is is 2n+3, I can't figure it out :/ – Alpha Aug 28 '16 at 23:20
  • @Alpha First, what is $T_{(n+1)}$? HINT: Replace "$n$" with "$n+1$" in $n(n+2)$ . . . – Noah Schweber Aug 28 '16 at 23:22
  • I answered above to your question about students, dunno if you read it but just making sure maybe it'll help you – Alpha Aug 28 '16 at 23:41
  • again, thank you for help – Alpha Aug 28 '16 at 23:42