7, 9, 12, ?, 24, 36, 56, 90

Here was a problem shown to me by one of my students which he picked up in Oxford on an open day. It has a surprising solution so I thought I'd give it a write up. Which number should replace the ? In the sequence 7, 9, 12, ?, 24, 36, 56, 90? In case you want to have a go I'll put my answer below. It is worth a little play around with.

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This is a sequence designed by the ever brilliant Roger Penrose. The missing term is 24ln2 (where ln is the natural logarithm which comes up in the second year of A Level maths). The general formula is 24(2^n-1)/n where n=-3, -2, -1, 1, 2, 3 and 4. The missing value is the value when n=0. There was a reason that we only have the terms from 7 to 90, because the terms just before (45/8) and after (744/5) are non integer although there are occasional integer values further along the sequence in each direction.

Ok, but how do we get logarithms involved? Popping zero into the formula we get 24(0)/0 which is undefined. Instead we are going to use L’Hopital’s Rule (explained in this video post). Here's the working:

I'm going to gloss over the details because the video link above spells it out, but the gist is that if we differentiate the top and the bottom and their limit is defined, then their limit is the same as the original limit. To differentiate 2^n we have to take ln of both sides and differentiate implicitly which I do on the right hand side of the page.