An arithmetic sequence whose first term is 2 has the property that its second, third, and seventh terms are consecutive terms of a geometric sequence. Determine all possible second terms of the arithmetic sequence.

Step 1 Given: First of an arithmetic sequence is 2 and has the property that its second, third, and seventh terms are consecutive terms of a geometric sequence. Step 2 To determine all possible second terms of an arithmetic sequence. Let, a be the first term of the arithmetic sequence. $\Rightarrow a=2.$ And d be the common difference. So, the terms of the of the arithmetic sequence will be: $a,a+d,a+2d,a+3d,...$ $\Rightarrow 2,2+d,2+2d,2+3d,....----.:a=2$ That is, to find all possible second terms of the arithmetic sequence. We need to find the value of d. Step 3 We have that second, third, and seventh terms of an arithmetic sequence are consecutive terms of a geometric sequence. $\Rightarrow \frac{a+2d}{a+d}=\frac{a+6d}{a+2d}$
$\Rightarrow (2+2d{)}^2=(2+6d)(2+d)$
$\Rightarrow 4+8d+4d^2=4+2d+12d+d^2$
$\Rightarrow 3d^2=6d$
$\Rightarrow d=2$ And second term of an arithmetic sequence is given by, $2+d=2+2=4.$ Therefore, the second of the arithmetic sequence is 4. Hence, an arithmetic sequence with the first term as 2 and with common ratio as 2 is given by 2, 4, 6, 8,.....