Question

which table represents an arithmetic sequence?

table 1:
\\(n\\) | 1 | 2 | 3 | 4 | 5
\\(a_n\\) | -6 | -12 | -24 | -48 | -96

table 2:
\\(n\\) | 1 | 2 | 3 | 4 | 5
\\(a_n\\) | 7 | 9 | 13 | 19 | 27

table 3:
\\(n\\) | 1 | 2 | 3 | 4 | 5
\\(a_n\\) | 8.7 | 7.3 | 5.9 | 4.5 | 3.1

the arithmetic sequence from the previous problem is 8.7, 7.3, 5.9, 4.5, 3.1, ... what is the common difference, \\(d\\), of this sequence?
\\(d = \\)

Explanation:

Identify the arithmetic sequence table

Using the Arithmetic Sequences knowledge point

• Table 1: The terms are \(-6, -12, -24, -48, -96\). The ratio between consecutive terms is constant (\(r = 2\)), representing a geometric sequence.
• Table 2: The terms are \(7, 9, 13, 19, 27\). The differences are \(2, 4, 6, 8\), which are not constant.
• Table 3: The terms are \(8.7, 7.3, 5.9, 4.5, 3.1\). The differences are:

$$ 7.3 - 8.7 = -1.4 $$

$$ 5.9 - 7.3 = -1.4 $$

$$ 4.5 - 5.9 = -1.4 $$

$$ 3.1 - 4.5 = -1.4 $$

Since the difference is constant, this table represents an arithmetic sequence.

Calculate the common difference

Using the Arithmetic Sequences knowledge point

• The sequence is \(8.7, 7.3, 5.9, 4.5, 3.1, \dots\)
• Calculate the common difference \(d\) by subtracting the first term from the second term:

$$ d = 7.3 - 8.7 = -1.4 $$

Answer:

Question 1

The table that represents an arithmetic sequence is the third table:

\(n\)	1	2	3	4	5

Question 2

\(d =\) <blank>-1.4</blank>