# 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.

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.
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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. $⇒a=2.$ And d be the common difference. So, the terms of the of the arithmetic sequence will be: $a,a+d,a+2d,a+3d,...$ $⇒2,2+d,2+2d,2+3d,....----.:a=2$ 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. $⇒\frac{a+2d}{a+d}=\frac{a+6d}{a+2d}$
$⇒\left(2+2d{\right)}^{2}=\left(2+6d\right)\left(2+d\right)$
$⇒4+8d+4{d}^{2}=4+2d+12d+{d}^{2}$
$⇒3{d}^{2}=6d$
$⇒d=2$ 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,.....