Question

which sequences are arithmetic? select three options.

\\(\square\\) \\(-8.6, -5.0, -1.4, 2.2, 5.8, \dots\\)

\\(\square\\) \\(2, -2.2, 2.42, -2.662, 2.9282, \dots\\)

\\(\square\\) \\(5, 1, -3, -7, -11, \dots\\)

\\(\square\\) \\(-3, 3, 9, 15, 21, \dots\\)

\\(\square\\) \\(-6.2, -3.1, -1.55, -0.775, -0.3875, \dots\\)

Explanation:

Define arithmetic sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference, denoted as \(d\). We can check if a sequence is arithmetic by calculating the difference between successive terms:

$$d = a_{n} - a_{n-1}$$

If this difference is the same for all consecutive pairs, the sequence is arithmetic.

Analyze the first sequence

Let's examine the first sequence: \(-8.6, -5.0, -1.4, 2.2, 5.8, \dots\)
Calculate the differences between consecutive terms:

$$-5.0 - (-8.6) = -5.0 + 8.6 = 3.6$$

$$-1.4 - (-5.0) = -1.4 + 5.0 = 3.6$$

$$2.2 - (-1.4) = 2.2 + 1.4 = 3.6$$

$$5.8 - 2.2 = 3.6$$

Since the difference is constantly \(3.6\), this is an arithmetic sequence.

Analyze the second sequence

Let's examine the second sequence: \(2, -2.2, 2.42, -2.662, 2.9282, \dots\)
Calculate the differences between consecutive terms:

$$-2.2 - 2 = -4.2$$

$$2.42 - (-2.2) = 4.62$$

Since the differences are not equal (\(-4.2
eq 4.62\)), this is not an arithmetic sequence. (It is a geometric sequence with a common ratio of \(-1.1\)).

Analyze the third sequence

Let's examine the third sequence: \(5, 1, -3, -7, -11, \dots\)
Calculate the differences between consecutive terms:

$$1 - 5 = -4$$

$$-3 - 1 = -4$$

$$-7 - (-3) = -4$$

$$-11 - (-7) = -4$$

Since the difference is constantly \(-4\), this is an arithmetic sequence.

Analyze the fourth and fifth sequences

Let's examine the fourth sequence: \(-3, 3, 9, 15, 21, \dots\)
Calculate the differences:

$$3 - (-3) = 6$$

$$9 - 3 = 6$$

$$15 - 9 = 6$$

$$21 - 15 = 6$$

Since the difference is constantly \(6\), this is an arithmetic sequence.

Let's examine the fifth sequence: \(-6.2, -3.1, -1.5, -0.775, -0.3875, \dots\)
Calculate the differences:

$$-3.1 - (-6.2) = 3.1$$

$$-1.55 - (-3.1) = 1.55$$

Since the differences are not equal, this is not an arithmetic sequence. (It is a geometric sequence with a common ratio of \(0.5\)).

Answer:

• (A) \(-8.6, -5.0, -1.4, 2.2, 5.8, \dots\) (Correct answer)
• (B) \(2, -2.2, 2.42, -2.662, 2.9282, \dots\)
• (C) \(5, 1, -3, -7, -11, \dots\) (Correct answer)
• (D) \(-3, 3, 9, 15, 21, \dots\) (Correct answer)
• (E) \(-6.2, -3.1, -1.55, -0.775, -0.3875, \dots\)