Question

What is the answer of 1/2 of 720?

Answer 1

The question is to guess a function f such that f(1) = 1, f(2) = 2, f(3) = 6, f(4) = 24, f(5) = 120 and f(6) = 720, and then compute f(7) to predict the next number in the sequence. An ‘obvious’ guess is f(x) = x!, as the other answers point out. This function will produce f(7) = 7! = 5040.

But it there are many more functions that fit the given data. For instance the following polynomial:

g(x) = -264 + (3059/5) x - (6139/12) x^2 + (1589/8) x^3 - (437/12) x^4 + (103/40) x^5

This has:

g(1) = 1
g(2) = 2
g(3) = 6
g(4) = 24
g(5) = 120
g(6) = 720
g(7) = 2921

And so the next number in the sequence could be 2921 instead of 5040. From the data alone, we cannot know which of the two functions, the factorial or the polynomial, was used to produce the data. There are many more functions that fit the data, so it could be neither! And so we cannot know for sure what will come after the initial sequence: 5040, 2921, or something else.

From the fact that these type of questions are usually after an`obvious’ answer, we can conclude that 5040 is probably the expected answer. It is worth thinking about why x! is more obvious than the polynomial g(x) though!






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