Decoding the Pattern: Determine the Term-to-Term Rule of 13, 16, 19, 22

Q: What's the difference between term-to-term rule and nth term formula?

The term-to-term rule tells you how to get from one term to the next (like +3). The nth term formula lets you find any term directly without calculating all the previous ones!

Q: How do I know if I found the right common difference?

Check that the difference is the same between every pair of consecutive terms. In this sequence: 16-13=3, 19-16=3, 22-19=3. All equal 3, so d=3 is correct!

Q: Why does the formula become 3n + 10 instead of staying as 13 + (n-1)×3?

Both forms are correct! We expand $ 13 + (n-1) \times 3 $ to get $ 13 + 3n - 3 = 3n + 10 $. The simplified form is easier to use.

Q: Can I check my answer by plugging in different values of n?

Absolutely! Test your formula: For n=1: 3(1)+10=13 ✓, n=2: 3(2)+10=16 ✓. If all match the original sequence, your formula is correct!

Q: What if the sequence doesn't have a constant difference?

Then it's not an arithmetic sequence! Check for other patterns like multiplying by a constant (geometric) or look for quadratic patterns if differences of differences are constant.

Arithmetic Sequences with Nth Term Formula

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

13, 16, 19, 22

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the sequence formula

00:04 Identify the first term according to the given data

00:11 Note the constant difference between terms

00:17 This is the constant difference

00:21 Use the formula to describe an arithmetic sequence

00:26 Substitute appropriate values and solve to find the sequence formula

00:38 Properly expand brackets, multiply by each factor

00:47 Continue solving

00:57 And this is the solution to the question

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?

13, 16, 19, 22

Step-by-step solution

To solve this problem, the sequence 13, 16, 19, 22 must be analyzed to find the term-to-term rule.

Step 1: Identify the common difference.

Subtract the first term from the second term:
$16 - 13 = 3$

Verify the difference throughout the sequence:
$19 - 16 = 3$
$22 - 19 = 3$

The common difference $d$ is $3$ .

Step 2: Use the arithmetic sequence formula.

Start with the general formula for an arithmetic sequence:
$a_n = a_1 + (n-1) \cdot d$

In this sequence, $a_1 = 13$ and $d = 3$ . Substitute these values in:
$a_n = 13 + (n-1) \cdot 3$

Simplify the expression:

$a_n = 13 + 3n - 3$

$a_n = 3n + 10$

The term-to-term rule of the sequence is $3n + 10$ .

Examining the given choices, the correct choice is:

$3n + 10$ (Choice 2).

Final Answer

$3n+10$

Key Points to Remember

Essential concepts to master this topic

Common Difference: Subtract consecutive terms to find constant difference
Formula: Use $a_n = a_1 + (n-1)d$ where $a_1 = 13$ and $d = 3$
Check: Verify $3n + 10$ gives 13, 16, 19, 22 for n = 1, 2, 3, 4 ✓

Common Mistakes

Avoid these frequent errors

Using the wrong formula or confusing term-to-term vs nth term
Don't write the common difference as the rule = just gives +3! This only tells how to get the next term, not the formula for any term. Always use the nth term formula $a_n = a_1 + (n-1)d$ to find the general rule.

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 rule and nth term formula?

The term-to-term rule tells you how to get from one term to the next (like +3). The nth term formula lets you find any term directly without calculating all the previous ones!

How do I know if I found the right common difference?

Check that the difference is the same between every pair of consecutive terms. In this sequence: 16-13=3, 19-16=3, 22-19=3. All equal 3, so d=3 is correct!

Why does the formula become 3n + 10 instead of staying as 13 + (n-1)×3?

Both forms are correct! We expand $13 + (n-1) \times 3$ to get $13 + 3n - 3 = 3n + 10$ . The simplified form is easier to use.

Can I check my answer by plugging in different values of n?

Absolutely! Test your formula: For n=1: 3(1)+10=13 ✓, n=2: 3(2)+10=16 ✓. If all match the original sequence, your formula is correct!

What if the sequence doesn't have a constant difference?

Then it's not an arithmetic sequence! Check for other patterns like multiplying by a constant (geometric) or look for quadratic patterns if differences of differences are constant.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Series questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime