Problem

(2) Identify each sequence as arithmetic or geometric. Then determine the common difference or common ratio for each sequence.
(a) $2,5,8,11,14,17$
(b) $1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{1}{256}$

Answer

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Answer

Final Answer: (b) The sequence \(1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{1}{256}\) is a geometric sequence with a common ratio of \(\boxed{0.25}\).

Steps

Step 1 :Identify each sequence as arithmetic or geometric. An arithmetic sequence is one in which the difference between each pair of consecutive terms is constant. A geometric sequence is one in which the ratio of each pair of consecutive terms is constant.

Step 2 :For sequence (a), subtract the second term from the first term, the third term from the second term, and so on, to see if the differences are constant.

Step 3 :For sequence (b), divide the second term by the first term, the third term by the second term, and so on, to see if the ratios are constant.

Step 4 :From the calculations, we can see that sequence (a) is an arithmetic sequence with a common difference of 3, and sequence (b) is a geometric sequence with a common ratio of 0.25.

Step 5 :Final Answer: (a) The sequence \(2,5,8,11,14,17\) is an arithmetic sequence with a common difference of \(\boxed{3}\).

Step 6 :Final Answer: (b) The sequence \(1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{1}{256}\) is a geometric sequence with a common ratio of \(\boxed{0.25}\).

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