How Do you find the nth term 15 12 9 6?

Answer 1

There are infinitely many formula that give the four terms {15, 12, 9, 6} for n = 1, 2, 3, 4; the way they continue after n=4 will be different.

The nth term is given by:

U{n} = (13n⁴ - 130n³ + 455n² - 674n + 456)/8

Which makes U{5} = 42.

However, I suspect that your teacher wants the much simpler formula for the Arithmetic Progression with the initial term of 15 and common difference of -3:

U{n} = 18 - 3n

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.
The simplest rule here, based on a linear expression, isT(n) = 18 - 3*n for n = 1, 2, 3, ...