My Math Forum Trying to figure out formula for nth term of this sequence

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 August 24th, 2015, 03:52 PM #1 Newbie   Joined: Jun 2009 Posts: 28 Thanks: 1 Trying to figure out formula for nth term of this sequence Hi, I was trying to use Wolfram Alpha to come up with a formula for the nth term of this sequence: 1/64, 3/128, 1/32, 3/64, 1/16, 3/32, 1/8, 3/16, 1/4, 3/8, 1/2, ... This is the URL for the formula that the tool generated: 1/64,3/128,1/32,3/64,1/16,3/32,1/8,3/16,1/4,3/8,1/2 - Wolfram|Alpha Results If I calculate the first term using that formula, it comes out to be -5/128. It should be 1/64. However, I ignored the G(n) x a(n), and just treated it like it was a(z). That was probably the problem. What do the G(n) and a(n) before the (z) mean, then? If it's some kind of composite function, shouldn't they have been more clear - as to what they mean? Thanks in advance. Andrew
August 24th, 2015, 05:08 PM   #2
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Hello, Jonas!

Quote:
 I was trying to come up with a formula for the nth term of this sequence: $\;\;\;\frac{1}{64},\;\frac{3}{128},\;\frac{1}{32}, \; \frac{3}{64},\;\frac{1}{16},\;\frac{3}{32},\;\frac {1}{8},\;\frac{3}{16},\;\frac{1}{4},\;\frac{3}{8}, \;\frac{1}{2}\;\cdots$

We have two sequences riffle-shuffled together.

$\;\;\;\;\;\begin{array}{c|c|c|c|c|c|c|c}
\text{odd }n & 1 & 3 & 5 & 7 & 9 & 11 & \cdots \\ \hline
& \frac{1}{64} & \frac{1}{32} & \frac{1}{16} & \frac{1}{8}& \frac{1}{4} & \frac{1}{2} & \cdots \end{array}$

$\;\;\;\begin{array}{c|c|c|c|c|c|c}
\text{even }n & 2 & 4 & 6 & 8 & 10 & \cdots \\ \hline
& \frac{3}{128} & \frac{3}{64} & \frac{3}{32} & \frac{3}{16} & \frac{3}{8} & \cdots \end{array}$

For odd $n$, the general term is: $\;\frac{1}{2^{\frac{13-n}{2}}}$
For even $n$, the general term is: $\;\frac{3}{2^{\frac{16-n}{2}}}$

$\text{Therefore: }\;f(n) \;=\;\left(\frac{1-(-1)^n}{2}\right)\cdot\frac{1}{2^{\frac{13-n}{2}}} \,+\, \left(\frac{1+(-1)^n}{2}\right)\cdot\frac{3}{2^{\frac{16-n}{2}}}$

 August 24th, 2015, 05:44 PM #3 Newbie   Joined: Jun 2009 Posts: 28 Thanks: 1 Wow, thanks! I need to develop my critical thinking/"figuring out" skills. Lol. Well, at least my memory is not abysmal. In fact, memorization is the gateway to all other cognitive skills. Without memorization, none of the other abilities would have any value. In order to think critically about something, one needs to have something retained, by way of memory, first. In other words, one must have something in one's mind to think critically about. Anyway, I have always wondered if there was a set way, or method, of looking at any sequence and deriving an nth term formula from it.
 August 26th, 2015, 06:06 PM #4 Newbie   Joined: Jun 2009 Posts: 28 Thanks: 1 Hm, that's weird. Every time I click on the URL link that I gave in my original post; the Wolfram Alpha tool shows different results. I clicked on it once, and it showed the actual formula for the nth term in closed form. I clicked on the link again, and it didn't show the formula that time. Once again, I clicked on it, and that time it showed a point-plot graph, a histogram, and the mean, median and standard deviation for the values I incipiently gave. Yes, I know. I have no life. Lol.
July 19th, 2017, 05:28 PM   #5
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Quote:
 Originally Posted by jonas What do the G(n) and a(n) before the (z) mean, then?
It means that the Maclaurin series for the given generating function of z has coefficients $a_0$, $a_1$, etc., which are the terms of your sequence.

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