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

00:10 Let's find the sequence formula.

00:16 Here is the first term based on the data given.

00:25 Notice how each term changes; this is called D.

00:37 D is the constant difference between terms.

00:41 Now, let's apply the formula for our sequence.

00:49 Substitute the values in, solve, and find our formula.

01:02 Great job! That's the solution to our 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

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

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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