What is the nth term formula for 2 15 34 59 90?

Answer 1

One possible solution is:

T{n} = 3x² + 4x - 5 for n = 1, 2, ..., 5

There is no guarantee this will work for any n > 5; formulae can be found which will depend upon what T{6, 7, ...} are.

For the above, the next term (n = 6) is T{6} = 127.

Another possible solution is:

T{n} = (-17x⁵ + 255x⁴ - 1445x³ + 3897x² - 4562x + 1920)/24

For n = 1, 2, 3, 4, 5; t{n} = 2, 15, 34, 59, 90

The next term this time (n = 6) is: T{6} = 42.

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.

The simplest rule is based on a quadratic and is

T(n) = 3*n^2 + 4*n - 5 for n = 1, 2, 3, ...