Write three numbers between 2 and 22 so that the five numbers form an arithmetic sequence.

Step-by-step solution

1. 1

   Count the terms, not the inserted numbers. Three numbers inserted between two given ones creates a sequence of $3+2=5$ terms, with $u_1=2$ and $u_5=22$.

2. 2

   Set up the nth-term equation. $u_5=u_1+(5-1)d\quad\Longrightarrow\quad 22=2+4d$

3. 3

   Solve for the common difference. $4d=20\quad\Longrightarrow\quad d=5$

4. 4

   Build the three means. $u_2=2+5=7$, $u_3=7+5=12$, $u_4=12+5=17$.

5. 5

   Verify the completed run. $2,7,12,17,22$ has constant gaps of $5$ and ends exactly at $22$, so the answer is C. Option A has gap $6$ from $2$ to $6$ but $4$ from $18$ to $22$, so it is not arithmetic.