**Guest post by Luke Westbrook**

Do you remember back in high school, even so far back as elementary school, taking those standardized tests? Inevitably, in the math portion of whatever test, there would be at least one question in which the first few numbers of a sequence were shown, and you had to determine what the next number was in the sequence. Remember those?

It’s been quite some time since I’ve taken one of those tests, but I don’t remember being particularly intimidated by them. You just had to figure out the pattern. Simple. Well, in honor of those “simple” standardized test questions, we decided to test your pattern recognition. But I bet you can’t guess the pattern this time.

Here it is. Determine, if you dare, the next number of the following sequence:

1, 3, 5, 7, …

Pretty easy, right? I mean, come on! It’s so obvious. The next number is *clearly* 217,341.

If you’re confused, good. You know, in retrospect, this would have been a good blog for April Fool’s Day. Oh well.

Allow me to explain, using PTC Mathcad, of course.

You see, the sequence given above is not consecutive odd numbers, as it would appear to be, but rather the outputs of consecutive integer inputs of the following equation:

If we evaluate the consecutive integer inputs, we get:

And just for kicks, let’s look at the plot of this function.

So there you go. The proper progression to the sequence given above is

1, 3, 5, 7, 217341, 1086671, 3259993, …

Still don’t believe me? Try it out for free with PTC Mathcad Express.

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Better in this form:

f(x):=(x-x1)*(x-x2)*(x-x3)*(x-x4)*(x-x5)*(x-x6)*(x-x7)*(x-x8)*(x-x9)…

Hah! I *knew* they were trying to trick me when I did my first IQ test!

The test had a sequence very much like that at the start and I spent ages trying to think of the Correct Very Clever Answer; after all, I reasoned, it was an intelligence test and “9” was such a no-brainer that it just couldn’t be the answer… I didn’t do very well on that test. 😦

However, as an aside, I think it would be useful to extend Mathcad range definitions to go beyond simply numeric linear sequences and allow the encoding of polynomial numbers and string sequences as well. For example,

“a”..”e” would generate the sequence “a”,”b”,”c”,”d”,”e”

“a”,”c”..”e” would generate the sequence “a”,”c”,”e”

4,11,22..80 would generate the sequence 4,11,22,37,56,79 (1+x+2x^2)