Determine the Term-to-Term Rule for the Sequence: -1, 3, 7

Q: What's the difference between term-to-term and position-to-term rules?

Term-to-term tells you how to get from one term to the next (like 'add 4'). Position-to-term gives you a formula to find any term directly using its position, like $ a_n = 4n - 5 $.

Q: Why do I need to check if the common difference is the same?

If the differences aren't equal, it's not an arithmetic sequence! You need a consistent pattern to use the arithmetic sequence formula $ a_n = a_1 + (n-1)d $.

Q: How do I remember which formula to use?

For arithmetic sequences, always use $ a_n = a_1 + (n-1)d $. Think: start with first term, then add the common difference (n-1) times to reach position n.

Q: What if I get confused with the algebra in the final step?

Take it step by step! $ a_n = -1 + (n-1) \times 4 $ becomes $ a_n = -1 + 4n - 4 $, then combine like terms: $ a_n = 4n - 5 $.

Q: Can I use this method for any arithmetic sequence?

Yes! As long as you have a constant common difference, this formula works. Just substitute your specific values for $ a_1 $ and $ d $.

Arithmetic Sequences with Linear Term Rules

What is the term-to-term rule for the sequence below?

$-1,3,7$

❤️ 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

00:09 Notice the constant difference between terms

00:14 This is the constant difference

00:18 Use the formula to describe an arithmetic sequence

00:22 Substitute appropriate values and solve to find the sequence formula

00:39 Open parentheses properly, multiply by each factor

00:43 Continue solving

00:54 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 for the sequence below?

$-1,3,7$

Step-by-step solution

To solve this problem, we'll determine the term-to-term rule by identifying if this sequence is arithmetic and calculating the common difference.

Step 1: Calculate the common difference $d$ .
From the given sequence, compute $d = a_2 - a_1 = 3 - (-1) = 4$ ; similarly, $d = a_3 - a_2 = 7 - 3 = 4$ .
Step 2: Formulate the general term $a_n$ of the sequence.
Since the sequence has a common difference of $4$ , it is an arithmetic sequence. The formula for an arithmetic sequence is given by $a_n = a_1 + (n-1)d$ .
Step 3: Substitute the known values and simplify.
Using $a_1 = -1$ and $d = 4$ , the expression becomes $a_n = -1 + (n-1) \times 4$ which simplifies to $a_n = -1 + 4n - 4 = 4n - 5$ .
Step 4: Verify the formula with the given terms.
Check $a_1 = 4 \times 1 - 5 = -1$ ; $a_2 = 4 \times 2 - 5 = 3$ ; $a_3 = 4 \times 3 - 5 = 7$ . All match the given sequence.

Therefore, the term-to-term rule for the sequence is $4n - 5$ .

Among the choices provided, the correct option is :

$4n-5$

Final Answer

$4n-5$

Key Points to Remember

Essential concepts to master this topic

Common Difference: Calculate d = next term - previous term consistently
Formula Method: Use $a_n = a_1 + (n-1)d$ where $a_1 = -1$ and $d = 4$
Verification: Check each term: $4(1)-5 = -1$ , $4(2)-5 = 3$ , $4(3)-5 = 7$ ✓

Common Mistakes

Avoid these frequent errors

Using position-to-term rule instead of finding nth term formula
Don't just say 'add 4 each time' = incomplete answer! Position-to-term rules tell you how to get the next term, but the question asks for the nth term formula. Always find the general formula $a_n = 4n - 5$ that works for any position n.

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 and position-to-term rules?

Term-to-term tells you how to get from one term to the next (like 'add 4'). Position-to-term gives you a formula to find any term directly using its position, like $a_n = 4n - 5$ .

Why do I need to check if the common difference is the same?

If the differences aren't equal, it's not an arithmetic sequence! You need a consistent pattern to use the arithmetic sequence formula $a_n = a_1 + (n-1)d$ .

How do I remember which formula to use?

For arithmetic sequences, always use $a_n = a_1 + (n-1)d$ . Think: start with first term, then add the common difference (n-1) times to reach position n.

What if I get confused with the algebra in the final step?

Take it step by step! $a_n = -1 + (n-1) \times 4$ becomes $a_n = -1 + 4n - 4$ , then combine like terms: $a_n = 4n - 5$ .

Can I use this method for any arithmetic sequence?

Yes! As long as you have a constant common difference, this formula works. Just substitute your specific values for $a_1$ and $d$ .

🌟 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