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.

Answer 1

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}]

Answer 2

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

Answer 3

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

Answer 4

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