2008-II-43

Ans: B
I must be an arithmetic sequence.

$\begin{array}{rcl}
(2a+7) – (a+10) & = & a – 3 \\
(3a+4) – ( 2a+7) & = & a-3 \\
(4a+1) – ( 3a+4) & = & a-3
\end{array}$

There is a common difference $a-3$.

II may not be an arithmetic sequence. Suppose $b=1$, then the four terms become $7$, $62$, $509$ and $4092$.

$\begin{array}{rcl}
62 – 7 & = & 55 \\
509 – 62 & = & 447
\end{array}$

There is no common difference.

III must be an arithmetic sequence.

$\begin{array}{rcl}
\log c^8 – \log c^3 & = & 8\log c – 3 \log c \\
& = & 5 \log c \\
\log c^{13} – \log c^8 & = & 13 \log c – 8 \log c \\
& = & 5 \log c \\
\log c^{18} – \log c^{13} & = & 18 \log c – 13 \log c \\
& = & 5 \log c
\end{array}$

There is a common difference $5 \log c$.