Question 1

Question

A)
Which of the following statements correctly describes the sequence 3, 8, 13, 18, . . .?

Statement #1: It is an arithmetic sequence.

Statement #2: It is a geometric sequence.

(1 point)
Statement #$$ correctly describes the sequence.
Question 2
A)
Classify the following set of numbers by determining which of the options is true.

3, 1, 13, 19, 127, . . .

Option #1: The set is an arithmetic sequence.

Option #2: The set is a geometric sequence.

Option #3: The set is some other type of sequence.

Option #4: The set is not a sequence.

(1 point)
The true statement is Option #$$.
Question 3
A)
What is the common difference in the following arithmetic sequence?

34, 64, 94, . . .

(1 point)
The common difference in the arithmetic sequence is $$.
Question 4
A)
Given the first five terms of the sequence 3, 7, 11, 15, 19 . . . , determine which option is the correct subscript notation for the explicit formula.

Option #1: an=3+4n

Option #2: an=4+3n

Option #3: an=−1+4n

(1 point)
Option #$$
Question 5
A)
Given the first five terms of the sequence an=2, 6, 18, 54, 162 . . . , determine which option is the correct subscript notation for the recursive formula.

Option #1: a1=2; an=3an−1

Option #2: a1=2; an=3an+1

Option #3: a1=2; an=3an−1+2

(1 point)
Option #$$
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Answer 1

Let's answer each of your questions one by one:

Question 1
The sequence given is 3, 8, 13, 18, ...

• The differences between consecutive terms are:

  • 8 - 3 = 5
  • 13 - 8 = 5
  • 18 - 13 = 5
    This shows that the sequence has a common difference of 5, making it an arithmetic sequence.

• Statement #1: It is an arithmetic sequence. (True)

• Statement #2: It is a geometric sequence. (False)

The correct answer is Statement #1.

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Question 2
The set of numbers given is 3, 1, 13, 19, 127, ...

• This sequence does not have a common difference (not arithmetic) and does not have a common ratio (not geometric). It appears to be an irregular sequence with distinct terms; hence, it could be classified as some other type of sequence.

The true statement is Option #3 (The set is some other type of sequence).

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Question 3
The arithmetic sequence given is 34, 64, 94, ...

• The common difference can be found by subtracting the first term from the second term:
  64 - 34 = 30
  So, the common difference in this arithmetic sequence is 30.

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Question 4
The sequence given is 3, 7, 11, 15, 19, ...

• This sequence has a common difference of 4. The first term (a1) is 3. The explicit formula for an arithmetic sequence can be written as:
  \( a_n = a_1 + (n - 1)d \)
  where \( d \) is the common difference. So in this case, it is:
  \( a_n = 3 + 4(n - 1) \)
  This simplifies to:
  \( a_n = -1 + 4n \)

Choosing from the given options, we see that it matches Option #3: \( a_n = -1 + 4n \).

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Question 5
The sequence given is 2, 6, 18, 54, 162, ...

• Each term is obtained by multiplying the previous term by 3, starting from \( a_1 = 2 \).
  The recursive formula can be expressed as:
• \( a_1 = 2; a_n = 3a_{n-1} \)

Thus, the correct option here is Option #1: a1=2; an=3an−1.

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Summarizing the answers:

1. Statement #1
2. Option #3
3. 30
4. Option #3
5. Option #1