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?

$-1,3,7$

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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

00:09 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

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

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 $d$ .

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