30.4. In an arithmetic sequence, the third, fifth and eleventh terms are distinct and form a geometric sequence. If the fourth term of the arithmetic sequence is 6 , what is its 2007 th term?
30.4. 6015
Let $a$ and $d$ be the first term and the common difference of the given arithmetic sequence. Since there are distinct terms of the sequence, it follows that $d \neq 0$. Since the third, fifth and eleventh terms form a geometric sequence, we have
$$
\frac{a+4 d}{a+2 d}=\frac{a+10 d}{a+4 d} \quad \text { or } \quad a+d=0 .
$$

Since the fourth term of the sequence is 6 , we also have
$$
a+3 d=6 .
$$

Solving the system of equations involving $(1 \star)$ and $(2 \star)$, we get $a=-3$ and $d=3$. Thus, the 2007 th term is $-3+2006(3)=6015$.