Question

13 multiple answer 1 point which of the following is an arithmetic sequence? 1/2, 3/4, 1, 5/4, 3/2, ... 4, 2, 1, 1/2, 1/4, ... 27, 9, 3, 1, 1/3, ... 3/7, 7/3, 3/7, 7/3, 3/7, ... 14 multiple choice 1 point which table represents values of x and y such that y = x + 7? x y -2 -9 7 0 x y -8 -1 0 7 x y -3 -10 7 0 x y -6 -4

Explanation:

Step1: Recall arithmetic - sequence definition

An arithmetic sequence has a common difference \(d=a_{n + 1}-a_{n}\).
For the sequence \(\frac{1}{2},\frac{3}{4},1,\frac{5}{4},\frac{3}{2},\cdots\):
\(a_1=\frac{1}{2}\), \(a_2 = \frac{3}{4}\), \(d=\frac{3}{4}-\frac{1}{2}=\frac{3 - 2}{4}=\frac{1}{4}\).
\(a_3 = 1\), \(1-\frac{3}{4}=\frac{1}{4}\), \(\frac{5}{4}-1=\frac{1}{4}\), \(\frac{3}{2}-\frac{5}{4}=\frac{6 - 5}{4}=\frac{1}{4}\). It is an arithmetic sequence.
For the sequence \(4,2,1,\frac{1}{2},\frac{1}{4},\cdots\):
\(\frac{a_{2}}{a_{1}}=\frac{2}{4}=\frac{1}{2}\), \(\frac{a_{3}}{a_{2}}=\frac{1}{2}\), it is a geometric - sequence (not arithmetic).
For the sequence \(27,9,3,1,\frac{1}{3},\cdots\):
\(\frac{a_{2}}{a_{1}}=\frac{9}{27}=\frac{1}{3}\), \(\frac{a_{3}}{a_{2}}=\frac{3}{9}=\frac{1}{3}\), it is a geometric - sequence (not arithmetic).
For the sequence \(\frac{3}{7},\frac{7}{3},\frac{3}{7},\frac{7}{3},\frac{3}{7},\cdots\):
There is no common difference. \(\frac{7}{3}-\frac{3}{7}=\frac{49 - 9}{21}=\frac{40}{21}\), \(\frac{3}{7}-\frac{7}{3}=\frac{9 - 49}{21}=-\frac{40}{21}\), not an arithmetic sequence.

Step2: Check the table for \(y=x + 7\)

For the first table:
When \(x=-2\), \(y=-2 + 7=5
eq - 9\); when \(x = 7\), \(y=7 + 7 = 14
eq0\).
For the second table:
When \(x=-8\), \(y=-8 + 7=-1\); when \(x = 0\), \(y=0+7 = 7\). This table satisfies \(y=x + 7\).
For the third table:
When \(x=-3\), \(y=-3 + 7 = 4
eq-10\); when \(x = 7\), \(y=7 + 7 = 14
eq0\).
For the fourth table:
When \(x=-6\), \(y=-6 + 7 = 1
eq-4\).

Answer:

For question 13: \(\frac{1}{2},\frac{3}{4},1,\frac{5}{4},\frac{3}{2},\cdots\)
For question 14: The table with \(x=-8,y=-1\) and \(x = 0,y = 7\)