Question

find the 12th term of the arithmetic progression 1/2, 1/6, 1/10

Answer 1

1/10 = 3/30
1/6 = 5/30
---------- subtract
d = -2/30 = -1/15

check it first two points
d = 5/30 - 15/30 = -10/30 = -1/3

This is not an arithmetic progression, typo.

Answer 2

halfway between 1/2 and 1/10
is
(.5 + .1)/2 = .3 = 3/10 not 1/6

Answer 3

However, if we consider the sequence as

1/2 * 1/1, 1/2 * 1/3, 1/2 * 1/5

Tn = (1/2) * 1/(2n-1)

T12 = (1/2)(1/23) = 1/46

There are other rules which would produce the given terms, but this is the easiest for me to see.