Identify the Term-to-Term Rule in the Arithmetic Sequence: 3, 7, 11, 15

What is the term-to-term rule of the following sequence?

3, 7, 11, 15, ...

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find the sequence formula
00:03 Identify the first term according to the given data
00:08 Note the constant difference between terms
00:15 This is the constant difference
00:19 Use the formula to describe an arithmetic sequence
00:24 Substitute appropriate values and solve to find the sequence formula
00:42 Open parentheses properly, multiply by each factor
00:50 Continue solving
00:57 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

What is the term-to-term rule of the following sequence?

3, 7, 11, 15, ...

2

Step-by-step solution

To solve this problem, we'll follow these steps:

  • Identify the common difference.
  • Use the arithmetic sequence formula to find a general rule.
  • Compare the rule with the given choices to identify the correct one.

Now, let's work through each step:
Step 1: Identify the common difference.
By examining the sequence: 3,7,11,15,3, 7, 11, 15, \ldots, we find that the difference (dd) between each pair of consecutive terms is 73=47 - 3 = 4, 117=411 - 7 = 4, and 1511=415 - 11 = 4. Hence, the common difference dd is 4.
Step 2: Use the arithmetic sequence formula.
The first term a1a_1 is 3. Using the formula for the nth term of an arithmetic sequence: an=a1+(n1)d a_n = a_1 + (n-1)d
Substitute a1=3a_1 = 3 and d=4d = 4:
an=3+(n1)4 a_n = 3 + (n-1) \cdot 4
Simplify this equation:
an=3+4n4 a_n = 3 + 4n - 4
an=4n1 a_n = 4n - 1
Step 3: Compare the rule with the provided choices.
The derived formula is an=4n1 a_n = 4n - 1 which matches the choice (2):4n1 \textbf{(2)}: 4n - 1 .

Therefore, the term-to-term rule of the sequence is 4n1\mathbf{4n - 1}.

3

Final Answer

4n1 4n-1

Practice Quiz

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Is there a term-to-term rule for the sequence below?

18 , 22 , 26 , 30

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