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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the sequence formula

00:04 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

12 ☐ 10 ☐ 8 7 6 5 4 3 2 1

Which numbers are missing from the sequence so that the sequence has a term-to-term rule?

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.

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