The center of math retweeted the following problem:
I surmised the answer is 22, using the following reasoning:
Odd entries increase by 2, whereas even entries increase by 1
$a_{1}=16, \, a_{2}=17, \, \dotso$
All seemed fine until I posed this question to a mathematician, who claimed this was not a mathematical sequence. Or at least it is not known to the mathematical community, i.e.
So, I am led to ask is this a sequence? Whether the answer turns out to be yes or no, I am more interested in the argument. It would be most helpful if there was some truth from the answer, which I could take away for future endeavors in the mathematical world.
A sequence is a mapping whose domain is a subset of the naturals.
So define $a: 0 \mapsto 16, 1 \mapsto 17, 2 \mapsto 18, 3 \mapsto 18, 4 \mapsto 20, 5 \mapsto 19$
There's your sequence. As to what the next number is, there's no way to know. I could say that the sequence continues as follows:
$a_n = 0$ for $n \ge 6$
There's no rule that a sequence has to follow any predictable pattern. And even if you figure out one pattern, the person who constructed these problems could have had another pattern in mind.
That's why these pattern-finding problems are at best not mathematics, and are at worse stupid.
Note that the sequence could be:
$a_n = \displaystyle -\frac {7}{90}n^5 + \frac {11}{9} n^4 - \frac {62}9 n^3 + \frac {148} 9 n^2 - \frac {401}{30} n + 16$
In which case the next term would be $- \dfrac {22}3$