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,.....

\(\Rightarrow (2 + 2d)^2 = (2 + 6d)(2+d)\)

\(\Rightarrow 4 + 8d + 4d^2 = 4 + 2d + 12d + d^2\)

\(\Rightarrow 3d^2 = 6d\)

\(\Rightarrow d = 2\) 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,.....