Find the Pattern Rule: Analyzing Sequence 3,15,27,39,41 Using Variable n

Q: Why does the formula use (n-1) instead of just n?

Because we want the first term when n=1! Using (n-1) means when n=1, we get (1-1)=0, so a(1) = 3 + 0×12 = 3, which matches our sequence.

Q: What about the number 41 in the sequence?

The number 41 doesn't follow the arithmetic pattern! The first four terms (3, 15, 27, 39) have a common difference of 12, but 41 breaks this pattern since 41-39=2, not 12.

Q: How do I find the common difference?

Subtract any term from the next term: $ d = a_2 - a_1 $. For example: 15-3=12. Always verify by checking other consecutive pairs: 27-15=12, 39-27=12.

Q: Can I use this formula to find any term in the sequence?

Yes! Once you have $ a(n) = 3 + (n-1) \times 12 $, you can find any term. For example, the 10th term: a(10) = 3 + (10-1)×12 = 3 + 108 = 111.

Q: What if I get confused about which term is which?

Always remember: n represents the position in the sequence. So n=1 gives the 1st term (3), n=2 gives the 2nd term (15), and so on. The formula adjusts for this with (n-1).

Arithmetic Sequences with Pattern Recognition

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

$3,15,27,39,41$

Describe the property using the variable $n$

❤️ 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

00:05 This is the first term according to the given data

00:10 Let's identify the difference between terms (D) according to the given data

00:22 We'll use the formula to describe the sequence

00:32 We'll substitute appropriate values and solve to find the sequence formula

00:41 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:

$3,15,27,39,41$

Describe the property using the variable $n$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Recognize the pattern in the series.
Step 2: Identify the first term and the common difference.
Step 3: Derive the formula for the sequence.

Now, let's work through each step:

Step 1: The given series is $3, 15, 27, 39, 41$ . Observing the first few terms, $3, 15, 27, 39$ , we note each subsequent number increases by $12$ , forming an arithmetic sequence. The number $41$ does not follow this arithmetic sequence pattern.

Step 2: The first term $a_1$ is $3$ , and the common difference $d$ is $12$ , as derived from verifying the difference between each two successive terms.

Step 3: We use the arithmetic sequence formula:
$\displaystyle a(n) = a_1 + (n-1) \times d$

Substitute the known values:
$\displaystyle a(n) = 3 + (n-1) \times 12$

This formula describes the arithmetic sequence for the original numbers $3, 15, 27, 39$ but not for $41$ .

Therefore, the solution to the problem is:

$a(n) = 3 + (n-1) \times 12$ .

Final Answer

$a(n)=3+(n-1)\times12$

Key Points to Remember

Essential concepts to master this topic

Formula: Use a(n) = first term + (n-1) × common difference
Technique: Find common difference: 15-3 = 12, verify 27-15 = 12
Check: Test formula with known terms: a(2) = 3+(2-1)×12 = 15 ✓

Common Mistakes

Avoid these frequent errors

Using n instead of (n-1) in the formula
Don't write a(n) = 3 + n×12 = wrong position values! This shifts every term incorrectly because n=1 gives 15 instead of 3. Always use (n-1) to ensure the first term appears when n=1.

Practice Quiz

Test your knowledge with interactive questions

Is there a term-to-term rule for the sequence below?

18 , 22 , 26 , 30

FAQ

Everything you need to know about this question

Why does the formula use (n-1) instead of just n?

Because we want the first term when n=1! Using (n-1) means when n=1, we get (1-1)=0, so a(1) = 3 + 0×12 = 3, which matches our sequence.

What about the number 41 in the sequence?

The number 41 doesn't follow the arithmetic pattern! The first four terms (3, 15, 27, 39) have a common difference of 12, but 41 breaks this pattern since 41-39=2, not 12.

How do I find the common difference?

Subtract any term from the next term: $d = a_2 - a_1$ . For example: 15-3=12. Always verify by checking other consecutive pairs: 27-15=12, 39-27=12.

Can I use this formula to find any term in the sequence?

Yes! Once you have $a(n) = 3 + (n-1) \times 12$ , you can find any term. For example, the 10th term: a(10) = 3 + (10-1)×12 = 3 + 108 = 111.

What if I get confused about which term is which?

Always remember: n represents the position in the sequence. So n=1 gives the 1st term (3), n=2 gives the 2nd term (15), and so on. The formula adjusts for this with (n-1).

🌟 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