Question

what is the nth term in the sequence 12,6,4,3,2,12/5

Answer 1

One can quickly see that it is neither arithmetic nor geometric in nature,
nor is it harmonic.
I took several levels of "differences" and found no common difference, so
no simple polynomial expresses it
Not even Wolfram could find a pattern:
https://www.wolframalpha.com/input/?i=pattern+%7B12,6,4,3,2,12%2F5%7D

I tried a "best fit" and got this:
https://www.wolframalpha.com/input/?i=best+fit+%7B12,6,4,3,2,12%2F5%7D

I give up

Answer 2

Consider the ratio of successive terms. These ratios are
1/2, 2/3, 3/4, 4/6, 6/5
It was looking pretty good till the last two...

Answer 3

In this case, may be it best to use interpolation.

In wolframalpha. c o m paste:

interpolate ( 1 , 12 ) , ( 2 , 6 ) , ( 3 , 4 ) , ( 4 , 3 ) , ( 5 , 2 ) , ( 6, 12 / 5 )

interpolating polynomial is:

x ^ 5 / 300 + x ^ 4 / 30 - 21 x ^ 3 / 20 + 43 x ^ 2 / 6 - 3113 x / 150 + 133 / 5

OR

1 / 300 ( x ^ 5 + 10 x ^ 4 - 315 x ^ 3 + 2150 x ^ 2 - 6226 x + 7980 )

Replace x with n.

an = 1 / 300 ( n ^ 5 + 10 n ^ 4 - 315 n ^ 3 + 2150 n ^ 2 - 6226 n + 7980 )