How Do We Solve These Tricky Number Series Problems?

[CWatters]

My son has just started learning about number series and has managed to do all of his homework except for two questions that have him and me stumped. To give you some idea of the level he's at most of the questions were simple number series.. eg Find the nth term in 9, 2, -5, -12 to which the answer is -7n+16.

The two questions we are stuck on are:

1. Homework Statement

Q3) What is the equation for the nth term of the series.. 7, 11, 14, 15
Q4) What are the next two number in the series 0, 1, 1, 2, 5, 8, ?, ?

Homework Equations

None

The Attempt at a Solution

Q3) Sorry if the formatting is off...
Original 7 11 14 15
1st Dif 4 3 1
2nd Dif -1 -2
3rd Diff -1

So it might be a cubic quadratic (third level difference -1) but there aren't enough given numbers to be certain and anyway he hasn't covered quadratic series yet let alone cubic quadratic.

Q4) Not a clue.

Photocopies of the questions were provided so no typos unless the teacher made them.

We're missing something but what?

 

[mfig]

The second question looks suspiciously like a Fibonacci sequence, which is recursively defined. Are you missing a 3 after the 2?

 

[fresh_42]

I have a construction rule for the first one, Q3).
If $n$ is odd, say $n=2k-1$, then build $Q_3(n) = 7k$.
If $n$ is even, say $n=2k$, then build $Q_3(n) = 7+2^{k+1}$.
The next numbers would then be 21, 23, 28,...

The point is: this is voodoo! For any number $N$ you can find a sequence, i.e. a construction rule for which $N$ is the following number to your given ones. It's nothing very serious and I wouldn't pay too much attention on it. If you answer: 346298 and 57 are the next numbers of Q4) no one can ever prove you're wrong. It's more a "guess what I mean" than a serious mathematical task.

 

[fresh_42]

The second question looks suspiciously like a Fibonacci sequence, which is recursively defined. Are you missing a 3 after the 2?

Brilliant idea! Now I have a construction rule for Q4), too.
Build the Fibonacci sequence $F_n = F_{n-1} +F_{n-2}$ and drop every 5th member although calculating with it.
See: One can always find something weird and declare it as a construction rule.

 

[CWatters]

The second question looks suspiciously like a Fibonacci sequence, which is recursively defined. Are you missing a 3 after the 2?

Yes that popped up when I googled the numbers but the problem sheet definitely hasn't got a 3 in the sequence.

I also googled the sequence in the other question 7 11 14 15 which turns up in an obscure list of "numbers with at least one 1 and one 2 in ternary (base 3) representation"...
http://oeis.org/A125293
7 in base 3 is 21
11 in base 3 is 102
14 in base 3 is 112
15 in base 3 is 120
but the question asks for the nth term rather than the next two numbers so I can't see how this is relevant if at all.

 

[CWatters]

See: One can always find something weird and declare it as a construction rule.

Will be interesting to see what answer the teacher gives.

 

[fresh_42]

Will be interesting to see what answer the teacher gives.

What's the next letter in "M T W T" or in "J F M A"?

 

[CWatters]

F and M.

Is that meant to be a hint about how to solve my questions?

 

[fresh_42]

F and M.

Is that meant to be a hint about how to solve my questions?

No. It's a joke. And it shows the arbitrariness of such exercises. Btw: most mathematicians have much more trouble to find F and M

 

[haruspex]

No. It's a joke. And it shows the arbitrariness of such exercises. Btw: most mathematicians have much more trouble to find F and M

Agreed, which is why one should expect that the shorter the given sequence the simpler the rule.
Certainly Q4 is a misprint for the Fibonacci sequence. Since we are only given 4 numbers in Q3 it should be even simpler, so I very strongly suspect another error. Maybe 7, 11, 14, 16?

 

[SammyS]

Q3: Perhaps these numbers are written in base eleven.

In base ten the sequence is: 7, 12, 15, 16, which could be a quadratic.

 

[fresh_42]

Q3: Perhaps these numbers are written in base eleven.

In base ten the sequence is: 7, 12, 15, 16, which could be a quadratic.

Only that $16_{11} = 17_{10}$.
Edit: Sorry, I was reading haruspex' corrected version ... (Where's my deletion button gone?)