Arithmetic Sequence Rule: Determine the Increment for 4, 14, 24, 34

Q: Why isn't the answer 4n + something since the first term is 4?

The coefficient in the formula comes from the common difference, not the first term! Since we add 10 each time, the coefficient is 10. The first term (4) helps us find the constant term.

Q: How do I know which formula to pick from the multiple choice?

Test each formula with the given terms! For example, try $ n = 1 $: does $ 10(1) - 6 = 4 $? Yes! Then try $ n = 2 $: does $ 10(2) - 6 = 14 $? Yes!

Q: What if I get confused about the arithmetic sequence formula?

Remember the pattern: $ a_n = a_1 + (n-1)d $. Here, $ a_1 $ is the first term and $ d $ is the common difference. When you expand this, you get the simplified form like $ 10n - 6 $.

Q: Why do we subtract 1 from n in the formula?

Because when $ n = 1 $ (first term), we don't add any common differences yet! We need zero steps of the common difference. For the second term, we take one step, so $ (n-1) = 1 $.

Q: Can I work backwards from the answer choices?

Absolutely! This is often faster. Substitute $ n = 1, 2, 3, 4 $ into each choice and see which one gives you 4, 14, 24, 34. The correct formula will match all the given terms.

Arithmetic Sequences with Linear Formula Derivation

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

4, 14, 24, 34, ...

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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

00:19 Use the formula to describe an arithmetic sequence

00:27 Substitute appropriate values and solve to find the sequence formula

00:50 Open parentheses properly, multiply by each factor

00:53 Continue solving

00:59 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 of the sequence below?

4, 14, 24, 34, ...

Step-by-step solution

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

Step 1: Identify the sequence progression
Step 2: Find the common difference
Step 3: Formulate the $n$ -th term rule

Now, let's work through each step:

Step 1: Analyze the sequence. The sequence provided is 4, 14, 24, 34, ...

Step 2: Calculate the common difference by subtracting successive terms:
$14 - 4 = 10$ , $24 - 14 = 10$ , and $34 - 24 = 10$ .
The common difference is 10.

Step 3: Use the common difference and the first term to find the formula:
The sequence is arithmetic with the first term $a_1 = 4$ and common difference $d = 10$ .
The formula for the $n$ -th term of an arithmetic sequence is given by:
$a_n = a_1 + (n-1) \cdot d$ .
Plug $a_1 = 4$ and $d = 10$ into the formula:
$a_n = 4 + (n-1) \cdot 10$
$a_n = 4 + 10n - 10$
Thus, $a_n = 10n - 6$ .

Therefore, the sequence can be represented by the term-to-term rule:
$10n - 6$ .

Final Answer

$10n-6$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Calculate differences between consecutive terms to find common difference
Formula Application: Use $a_n = a_1 + (n-1)d$ where $a_1 = 4$ and $d = 10$
Verification: Check formula works: $10(2) - 6 = 14$ matches second term ✓

Common Mistakes

Avoid these frequent errors

Using first term as coefficient instead of common difference
Don't use the first term 4 as the coefficient in 4n - something = wrong formula! The coefficient must be the common difference (10). Always use the pattern: coefficient = common difference, then adjust the constant.

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 isn't the answer 4n + something since the first term is 4?

The coefficient in the formula comes from the common difference, not the first term! Since we add 10 each time, the coefficient is 10. The first term (4) helps us find the constant term.

How do I know which formula to pick from the multiple choice?

Test each formula with the given terms! For example, try $n = 1$ : does $10(1) - 6 = 4$ ? Yes! Then try $n = 2$ : does $10(2) - 6 = 14$ ? Yes!

What if I get confused about the arithmetic sequence formula?

Remember the pattern: $a_n = a_1 + (n-1)d$ . Here, $a_1$ is the first term and $d$ is the common difference. When you expand this, you get the simplified form like $10n - 6$ .

Why do we subtract 1 from n in the formula?

Because when $n = 1$ (first term), we don't add any common differences yet! We need zero steps of the common difference. For the second term, we take one step, so $(n-1) = 1$ .

Can I work backwards from the answer choices?

Absolutely! This is often faster. Substitute $n = 1, 2, 3, 4$ into each choice and see which one gives you 4, 14, 24, 34. The correct formula will match all the given terms.

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