Find the Variable Expression n for Sequence 12, 18, 24, 30, 36

Q: Why do we use (n-1) instead of just n in the formula?

Because when n = 1, we want the first term (12), not 12 + 6 = 18. Using $ (n-1) $ makes the formula work: $ a(1) = 12 + (1-1) \times 6 = 12 $.

Q: How do I find the common difference?

Subtract any term from the next term: $ 18 - 12 = 6 $, $ 24 - 18 = 6 $. The difference should be the same between all consecutive pairs.

Q: What if I want to find the 10th term?

Use the formula with n = 10: $ a(10) = 12 + (10-1) \times 6 = 12 + 54 = 66 $. The pattern continues beyond the given terms!

Q: Can I check my formula with the given numbers?

Yes! Test each position: $ a(1) = 12 $, $ a(2) = 18 $, $ a(3) = 24 $. If your formula gives these values, it's correct.

Q: What makes this an arithmetic sequence?

The constant difference of 6 between consecutive terms. This equal spacing is what defines an arithmetic sequence, making the formula predictable.

Arithmetic Sequences with Variable Expressions

Given the series whose difference between two jumped numbers is constant:

$12,18,24,30,36$

Describe the property using the variable $n$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the sequence formula

00:06 This is the first term according to the given data

00:15 Let's observe the change between terms (D) according to the given data

00:27 This is the constant difference in the sequence (D)

00:30 Let's use the formula to describe the sequence

00:39 Let's substitute appropriate values and solve to find the sequence formula

00:52 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

Given the series whose difference between two jumped numbers is constant:

$12,18,24,30,36$

Describe the property using the variable $n$

Step-by-step solution

To solve this problem, we first recognize that the sequence $12, 18, 24, 30, 36$ is an arithmetic sequence.

Step 1: Identify the first term $a_1$ .
The first term $a_1$ of the sequence is $12$ .
Step 2: Determine the common difference $d$ .
The difference between consecutive terms is consistent: $18 - 12 = 6$ . Hence, the common difference $d = 6$ .
Step 3: Formulate the expression for the general term.
The general term of an arithmetic sequence is given by $a(n) = a_1 + (n-1) \times d$ .
Step 4: Substitute the identified values into the formula.
Substituting the known values, we get $a(n) = 12 + (n-1) \times 6$ .

Thus, the expression for the general term of the given sequence is $a(n) = 12 + (n-1) \times 6$ , which corresponds to choice 3.

Final Answer

$a(n)=12+(n-1)\times6$

Key Points to Remember

Essential concepts to master this topic

Formula: General term uses $a(n) = a_1 + (n-1) \times d$
Technique: First term is 12, common difference is 6
Check: Substitute values: $a(3) = 12 + (3-1) \times 6 = 24$ ✓

Common Mistakes

Avoid these frequent errors

Using n instead of (n-1) in the formula
Don't write $a(n) = 12 + n \times 6$ = gives wrong values! This shifts every term by one position. Always use $a(n) = 12 + (n-1) \times 6$ to match the position correctly.

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

Why do we use (n-1) instead of just n in the formula?

Because when n = 1, we want the first term (12), not 12 + 6 = 18. Using $(n-1)$ makes the formula work: $a(1) = 12 + (1-1) \times 6 = 12$ .

How do I find the common difference?

Subtract any term from the next term: $18 - 12 = 6$ , $24 - 18 = 6$ . The difference should be the same between all consecutive pairs.

What if I want to find the 10th term?

Use the formula with n = 10: $a(10) = 12 + (10-1) \times 6 = 12 + 54 = 66$ . The pattern continues beyond the given terms!

Can I check my formula with the given numbers?

Yes! Test each position: $a(1) = 12$ , $a(2) = 18$ , $a(3) = 24$ . If your formula gives these values, it's correct.

What makes this an arithmetic sequence?

The constant difference of 6 between consecutive terms. This equal spacing is what defines an arithmetic sequence, making the formula predictable.

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