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.

tinfoQ 2020-12-15 Answered
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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l1koV
Answered 2020-12-16 Author has 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. 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. a+2da+d=a+6da+2d
(2+2d)2=(2+6d)(2+d)
4+8d+4d2=4+2d+12d+d2
3d2=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,.....
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