The first four terms of a sequence are given. Determine whether these terms can be the terms of an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic or geometric, find the next term.(a) 9, −9, 9, −9, arithmeticgeometric neither arithmetic or geometricIf the sequence is arithmetic or geometric, find the next term. (If the sequence is neither, enter DNE.) (b) 5, 35, 65, 1, arithmeticgeometric neither arithmetic or geometricIf the sequence is arithmetic or geometric, find the next term. (If the sequence is neither, enter DNE.) (c) 2, −1, 12, 2, arithmeticgeometric neither arithmetic or geometricIf the sequence is arithmetic or geometric, find the next term. (If the sequence is neither, enter DNE.) (d) x − 1, x, x + 1, x + 2, arithmeticgeometric neither arithmetic or geometricIf the sequence is arithmetic or geometric, find the next term. (If the sequence is neither, enter DNE.)
Question
The first four terms of a sequence are given. Determine whether these terms can be the terms of an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic or geometric, find the next term.(a) 9, −9, 9, −9, arithmeticgeometric neither arithmetic or geometricIf the sequence is arithmetic or geometric, find the next term. (If the sequence is neither, enter DNE.) (b) 5, 35, 65, 1, arithmeticgeometric neither arithmetic or geometricIf the sequence is arithmetic or geometric, find the next term. (If the sequence is neither, enter DNE.) (c) 2, −1, 12, 2, arithmeticgeometric neither arithmetic or geometricIf the sequence is arithmetic or geometric, find the next term. (If the sequence is neither, enter DNE.) (d) x − 1, x, x + 1, x + 2, arithmeticgeometric neither arithmetic or geometricIf the sequence is arithmetic or geometric, find the next term. (If the sequence is neither, enter DNE.)
Solution
(a) The sequence is arithmetic because the difference between each term is constant (9 - (-9) = 18, -9 - 9 = -18). The next term would be -9 - 18 = -27.
(b) The sequence is neither arithmetic nor geometric because the difference between each term is not constant and the ratio of each term is not constant. Therefore, the next term does not exist (DNE).
(c) The sequence is neither arithmetic nor geometric because the difference between each term is not constant and the ratio of each term is not constant. Therefore, the next term does not exist (DNE).
(d) The sequence is arithmetic because the difference between each term is constant (x - (x - 1) = 1, x + 1 - x = 1, x + 2 - (x + 1) = 1). The next term would be x + 2 + 1 = x + 3.
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