What is the 8th term of 12 59 294 1469 7344?

Answer 1

There are an infinite number of possible answers - there are an infinite number of formulae that can be found to give t{1..5} = {12, 59, 294, 1469, 7344} which will give different values for t8.

eg:

t{n} = (376n⁴ - 3384n³ + 11186n² - 15369n + 7227)/3

gives t{1..5} = {12, 59, 294, 1469, 7344}, and t8 = 135889.

t{n} = (-4929n⁵ + 76943n⁴ - 446037n³ +1198513n² -1473498n + 649296)/24

also gives t{1..5} = {12, 59, 294, 1469, 7344}, but t8 = -381656.

However, the solution your teacher is probably expecting is based on the fact that:

U1 = 12

U{n} = 5U{n-1} - 1 for n ≥ 2

This leads to:

t1 = 12

t{n} = 12 + 47 × 5ⁿ⁻² for n ≥ 2

→ t8 = 917969

Answer 2

According to Wittgenstein's Finite Rule Paradox every finite sequence of numbers can be a described in infinitely many ways and so can be continued any of these ways - some simple, some complicated but all equally valid. Conversely, it is possible to find a rule such that any number of your choice can be the next one.

One possible solution can be obtained by fitting a polynomial of order 4.

Un = (376*n4 - 3384*n3 + 1186*n2 - 15369*n + 7227)/3 for n = 1, 2, 3, ...

which would give U8 = 135889.