tinfoQ

2020-12-15

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.

l1koV

Skilled2020-12-16Added 100 answers

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.
$\Rightarrow \frac{a+2d}{a+d}=\frac{a+6d}{a+2d}$

$\Rightarrow (2+2d{)}^{2}=(2+6d)(2+d)$

$\Rightarrow 4+8d+4{d}^{2}=4+2d+12d+{d}^{2}$

$\Rightarrow 3{d}^{2}=6d$

$\Rightarrow d=2$
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,.....

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