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

Q: What's the difference between term-to-term and position-to-term rules?

The term-to-term rule tells you how to get from one term to the next (like "add 4"). The position-to-term rule is a formula like $ 4n-1 $ that lets you find any term directly!

Q: How do I know if my common difference is right?

Check all consecutive pairs: 7-3=4, 11-7=4, 15-11=4. If you get the same number each time, that's your common difference!

Q: Why does the formula become 4n-1 instead of 3+4n?

Start with $ a_n = 3 + (n-1) \times 4 $. Expand: $ 3 + 4n - 4 $. Combine like terms: $ 4n - 1 $. Always simplify your final formula!

Q: Can I check my answer without the formula?

Yes! Test specific values: when n=3, does $ 4(3)-1 = 11 $? When n=4, does $ 4(4)-1 = 15 $? If both match the sequence, you're correct!

Q: What if I picked 4n+1 by mistake?

Test it! When n=1: $ 4(1)+1 = 5 $, but the first term is 3. This mismatch shows $ 4n+1 $ is wrong. Always verify with the actual sequence values!

Arithmetic Sequences with Linear Formula Derivation

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

00:07 Let's find the formula for this sequence.

00:11 First, identify the first term from the data.

00:15 Notice the constant difference between the terms.

00:22 This constant difference is important.

00:26 Now, let's use the formula for an arithmetic sequence.

00:31 Substitute the values into the formula, and solve to find it.

00:49 Open the parentheses carefully and multiply each part.

00:57 Keep going with the solution.

01:04 And that's how we solve this question. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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

3, 7, 11, 15, ...

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: $3, 7, 11, 15, \ldots$ , we find that the difference ( $d$ ) between each pair of consecutive terms is $7 - 3 = 4$ , $11 - 7 = 4$ , and $15 - 11 = 4$ . Hence, the common difference $d$ is 4.
Step 2: Use the arithmetic sequence formula.
The first term $a_1$ is 3. Using the formula for the nth term of an arithmetic sequence: $a_n = a_1 + (n-1)d$
Substitute $a_1 = 3$ and $d = 4$ :
$a_n = 3 + (n-1) \cdot 4$
Simplify this equation:
$a_n = 3 + 4n - 4$
$a_n = 4n - 1$
Step 3: Compare the rule with the provided choices.
The derived formula is $a_n = 4n - 1$ which matches the choice $\textbf{(2)}: 4n - 1$ .

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

Final Answer

$4n-1$

Key Points to Remember

Essential concepts to master this topic

Pattern: Find common difference by subtracting consecutive terms
Formula: Use $a_n = a_1 + (n-1)d$ where d = 4
Check: Verify $4n-1$ : n=1 gives 3, n=2 gives 7 ✓

Common Mistakes

Avoid these frequent errors

Confusing term-to-term rule with position-to-term formula
Don't think the common difference IS the formula = getting +4 instead of $4n-1$ ! The common difference tells you how terms change, but you need the full arithmetic sequence formula to find any term's value. Always use $a_n = a_1 + (n-1)d$ and simplify.

Practice Quiz

Test your knowledge with interactive questions

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 94,96,98,100,102,104 $$

FAQ

Everything you need to know about this question

What's the difference between term-to-term and position-to-term rules?

The term-to-term rule tells you how to get from one term to the next (like "add 4"). The position-to-term rule is a formula like $4n-1$ that lets you find any term directly!

How do I know if my common difference is right?

Check all consecutive pairs: 7-3=4, 11-7=4, 15-11=4. If you get the same number each time, that's your common difference!

Why does the formula become 4n-1 instead of 3+4n?

Start with $a_n = 3 + (n-1) \times 4$ . Expand: $3 + 4n - 4$ . Combine like terms: $4n - 1$ . Always simplify your final formula!

Can I check my answer without the formula?

Yes! Test specific values: when n=3, does $4(3)-1 = 11$ ? When n=4, does $4(4)-1 = 15$ ? If both match the sequence, you're correct!

What if I picked 4n+1 by mistake?

Test it! When n=1: $4(1)+1 = 5$ , but the first term is 3. This mismatch shows $4n+1$ is wrong. Always verify with the actual sequence values!

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