 # The 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. mattgondek4 2020-12-24 Answered
The 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.
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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 arithmetic sequence will be: $a,a+d,a+2d,a+3d,....⇒2,2d+d,2+2d,2+3d,...{:}^{\prime }a=2$ That is, to find all possible second terms of the arithmetic sequence. We need to find the value of d. 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,.....