Question

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.

Polynomial arithmetic
ANSWERED
asked 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.

Answers (1)

2020-12-16
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,.....
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