Quiz & Worksheet - Arithmetic Sequences | Study.com - Free Printable

Problem Analysis and Solution

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#### Question 1: Which of the following is not an arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference.

- Option A: \(1, 2, 3, 4, 5, \ldots\)
- Common difference: \(2 - 1 = 1\), \(3 - 2 = 1\), \(4 - 3 = 1\), etc.
- This is an arithmetic sequence with a common difference of \(1\).

- Option B: \(3, 9, 27, 81, \ldots\)
- Common difference: \(9 - 3 = 6\), \(27 - 9 = 18\), \(81 - 27 = 54\), etc.
- The differences are not constant. Instead, this is a geometric sequence where each term is multiplied by \(3\).
- This is not an arithmetic sequence.

- Option C: \(4.5, 5.0, 5.5, 6.0, \ldots\)
- Common difference: \(5.0 - 4.5 = 0.5\), \(5.5 - 5.0 = 0.5\), \(6.0 - 5.5 = 0.5\), etc.
- This is an arithmetic sequence with a common difference of \(0.5\).

- Option D: \(13, 2, -9, -20, -31, \ldots\)
- Common difference: \(2 - 13 = -11\), \(-9 - 2 = -11\), \(-20 - (-9) = -11\), etc.
- This is an arithmetic sequence with a common difference of \(-11\).

Answer to Question 1:
\(\boxed{B}\)

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#### Question 2: What is the 8th term of the following arithmetic sequence: \(-5, -1, 3, 7, 11, \ldots\)?

The general formula for the \(n\)-th term of an arithmetic sequence is:
\[
a_n = a_1 + (n-1)d
\]
where:
- \(a_n\) is the \(n\)-th term,
- \(a_1\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number.

From the given sequence:
- First term (\(a_1\)): \(-5\),
- Common difference (\(d\)): \(3 - (-1) = 4\).

We need to find the 8th term (\(a_8\)):
\[
a_8 = a_1 + (8-1)d
\]
\[
a_8 = -5 + 7 \cdot 4
\]
\[
a_8 = -5 + 28
\]
\[
a_8 = 23
\]

Answer to Question 2:
\(\boxed{23}\)

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#### Question 3: What is NOT true about the following arithmetic sequence: \(25, 32, 39, 46, \ldots\)?

From the given sequence:
- First term (\(a_1\)): \(25\),
- Common difference (\(d\)): \(32 - 25 = 7\).

Let's analyze each statement:

1. The common difference is 7.
- True. The difference between consecutive terms is \(7\).

2. The 7th term is 60.
- To find the 7th term (\(a_7\)):
\[
a_7 = a_1 + (7-1)d
\]
\[
a_7 = 25 + 6 \cdot 7
\]
\[
a_7 = 25 + 42
\]
\[
a_7 = 67
\]
- The 7th term is \(67\), not \(60\). This statement is false.

3. The 6th term is 60.
- To find the 6th term (\(a_6\)):
\[
a_6 = a_1 + (6-1)d
\]
\[
a_6 = 25 + 5 \cdot 7
\]
\[
a_6 = 25 + 35
\]
\[
a_6 = 60
\]
- The 6th term is \(60\). This statement is true.

4. The general rule is \(a(n) = 18 + 7n\).
- The general rule for an arithmetic sequence is:
\[
a_n = a_1 + (n-1)d
\]
Substituting \(a_1 = 25\) and \(d = 7\):
\[
a_n = 25 + (n-1) \cdot 7
\]
\[
a_n = 25 + 7n - 7
\]
\[
a_n = 18 + 7n
\]
- This statement is true.

Answer to Question 3:
\(\boxed{\text{The 7th term is 60.}}\)

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Final Answers:

1. \(\boxed{B}\)
2. \(\boxed{23}\)
3. \(\boxed{\text{The 7th term is 60.}}\)