find a formula to explain this set:$\{3,7,-11,-15,19,23,...\}$

Question

I want a formula to explain this set:$\{3,7,-11,-15,19,23,...\}$

Our teacher asked us to do.If they are one negative and one positive it was easy but now It become hard I think a lot, but no results.Please give me first some hints I want to solve it myself.

update1:the numbers are in the form of $4k-1$ without including negitive and positive.(two of them are negative and two of them are positive.

What are the next terms? My guess is: 42, 42, 42, 42, 42, 42, ... Am I correct? — Did, Jun 22 '16 at 08:46
And the terms from the 11th, are they 42, 42, 42, 42, 42, 42, 42, 42, ..., or not? — Did, Jun 22 '16 at 14:27
Not sure you got the point but here we go: your question would become suitable to this site only if you define mathematically the sequence. Until now, this is not done. — Did, Jun 22 '16 at 17:02

score 4 · Accepted Answer · answered Jun 22 '16 at 12:17

4

You're being asked to continue a sequence. I'm not going to get philosophical about the difference between sets and sequences, but to see it as a sequence helps us think of that invaluable math resource, the OEIS, a search engine for sequences of integers. I put in 3,7,-11,-15,19,23 and got http://oeis.org/A004767 a(n) = 4*n + 3. as the very first result.

Obviously $4n + 3$ does not account for the way the signs alternate, but other than that little detail, we're done. You already know that $(-1)^n$ alternates signs, but just a little too frequently for what you're trying to do. So you need a way to slow it down: $$(-1)^{\lfloor \frac{n}{2} \rfloor} (4n + 3).$$ Just to be absolutely sure, test it in Wolfram Alpha: Table[(-1)^(Floor[n/2])(4n + 3), {n, 0, 24}]

answered Jun 22 '16 at 12:17

Robert Soupe

14,663

where should I check in wolfram alpha. – Taha Akbari Jun 22 '16 at 12:47
Go to http://www.WolframAlpha.com and copy and paste that Table stuff Robert wrote, press Return. You can also use a looser syntax, e.g., (-1)^floor(n/2) * (4n + 3). – Mr. Brooks Jun 22 '16 at 21:06
Yes, what @Mr.Brooks said. With some caveats in mind, you will find Wolfram Alpha to be another invaluable resource for mathematics, especially if you don't have Wolfram Mathematica. – Robert Soupe Jun 23 '16 at 02:22

score 3 · Answer 2 · edited Jun 30 '16 at 21:36

3

$$ a_n = \left(\sin\left(\frac{n\pi}{2}\right) - \cos\left(\frac{n\pi}{2}\right)\right)(4n-1) $$

edited Jun 30 '16 at 21:36

Bob Happ

582

answered Jun 22 '16 at 05:27

boaz

4,783

oh what is this I want a simple one.I think It can be simpler.(surly) – Taha Akbari Jun 22 '16 at 05:28
this is simple (-: – boaz Jun 22 '16 at 05:30
I know we didn't learned trigonometry yet.but cool+1. – Taha Akbari Jun 22 '16 at 05:31
$(-1)^{(k^2+k+2)/2}(4k-1)$, then? – QTHalfTau Jun 22 '16 at 05:36
@QTHalfTau Yeah that's what I was referencing in my answer above (as they discuss triangular numbers in the link). – KR136 Jun 22 '16 at 05:39
@Αδριανός Ah should have checked that first; I was just getting rid of the trig for them. – QTHalfTau Jun 22 '16 at 05:44

score 3 · Answer 3 · edited Apr 13 '17 at 12:20

Notice the moduli of each element of the set follow an arithmetic sequence with common difference $4$. Hence you can derive the formula for a purely positive set.

But we intend for the set elements to alternate with a period of $2.$

The way you do this...it's is hard to provide a proper hint. To start of you know you'll need something of the form $(-1)^{f(x)}$ multiplied into your positive term formula.

You want a function which will be odd for 2 consecutive $x$ and then even for the next 2 consecutive $x$, and so on. Thus, you'll get your requisite pattern of positives and negatives. (Naturally minus one raised to an integer is negative when odd and positive when even)

If you're stumped, you can find the answer here: How to create alternating series with happening every two terms

score 3 · Answer 4 · answered Jun 22 '16 at 06:31

3

$${{a}_{n}}=\left\{ \begin{align} & -(4n-1)\,\,\,\,,\,\,\,\,\,\,\,\,n\overset{4}{\mathop{\equiv }}\,\,3\,\,\,,\,\,\,\,n\overset{4}{\mathop{\equiv }}\,\,0\, \\ & \,\,\,\,\,\,\,\,\,4n-1\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\operatorname{o}.w\, \\ \end{align} \right.$$

answered Jun 22 '16 at 06:31

Behrouz Maleki

11,182

these are two formulas I want a single one. – Taha Akbari Jun 22 '16 at 06:45
you didn't mention – Behrouz Maleki Jun 22 '16 at 06:48
.That isn't needed to mention you never write two formulas for a set. – Taha Akbari Jun 22 '16 at 06:50
why............? – Behrouz Maleki Jun 22 '16 at 06:52
A formula for a set contains three parts:first part for explaining the kind of the set second part for saying which set it belongs and third one to say it's range. – Taha Akbari Jun 22 '16 at 06:52
$$\sin \left( \frac{n\pi }{2} \right)-\cos \left( \frac{n\pi }{2} \right)=\left{ \begin{align} & 1,,,,,,,,,,,,,,,,?, \ & -1,,,,,,,,,,,,? \ \end{align} \right.$$ – Behrouz Maleki Jun 22 '16 at 06:57
Yeah it is right also add $4k-1$ – Taha Akbari Jun 22 '16 at 06:58

find a formula to explain this set:$\{3,7,-11,-15,19,23,...\}$

4 Answers4