Identify the Term-to-Term Rule: Analyzing the 8, 6, 4, 2 Sequence

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

The term-to-term rule tells you how to get from one term to the next (subtract 2). The position-to-term rule is a formula like $ -2n + 10 $ that gives you any term directly from its position.

Q: Why is the common difference negative?

The sequence is decreasing (8, 6, 4, 2...), so each term is smaller than the previous one. When you calculate 6 - 8 = -2, that negative sign shows the sequence goes down by 2 each time.

Q: How do I know which formula to use?

For arithmetic sequences (constant difference between terms), always use $ a_n = a_1 + (n-1)d $. Here, $ a_1 = 8 $ and $ d = -2 $.

Q: Can I check my answer with any term?

Yes! Try $ n = 3 $: $ -2(3) + 10 = -6 + 10 = 4 $. The 3rd term is indeed 4, so our formula $ -2n + 10 $ is correct!

Q: What if the sequence continued past 2?

Using our formula: the 5th term would be $ -2(5) + 10 = 0 $, the 6th term would be $ -2(6) + 10 = -2 $, and so on. The pattern continues infinitely!

Arithmetic Sequences with Negative Common Differences

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

8, 6, 4, 2, ...

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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

00:07 Note the constant difference between terms

00:13 This is the constant difference

00:18 Use the formula to describe an arithmetic sequence

00:24 Substitute appropriate values and solve to find the sequence formula

00:37 Expand brackets correctly, multiply by each factor

00:45 Continue solving

00:50 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 following sequence?

8, 6, 4, 2, ...

Step-by-step solution

To solve this problem, we'll approach it step by step:

Step 1: Identify the sequence pattern.
The sequence given is 8, 6, 4, 2, ... Each term is 2 less than the previous one.

Step 2: Determine the common difference.
The difference between any two consecutive terms (e.g., 6 - 8, 4 - 6) is $-2$ .

Step 3: Use the formula for the $n$ -th term of an arithmetic sequence.
For a sequence with first term $a_1 = 8$ and common difference $d = -2$ , the $n$ -th term can be calculated using:

$a_n = a_1 + (n-1) \cdot d$
This gives us:

$a_n = 8 + (n-1)(-2)$
$= 8 - 2n + 2$
$= -2n + 10$

Therefore, the rule for the sequence is $a_n = -2n + 10$ .

By comparing this with the given options, the correct choice is:

$-2n + 10$

Final Answer

$-2n+10$

Key Points to Remember

Essential concepts to master this topic

Pattern: Each term decreases by 2, so common difference is -2
Formula: Use $a_n = a_1 + (n-1)d$ where d = -2
Check: Verify n=1 gives 8: $-2(1) + 10 = 8$ ✓

Common Mistakes

Avoid these frequent errors

Confusing term-to-term rule with position-to-term rule
Don't say 'subtract 2' as the answer = incomplete understanding! The question asks for the nth term formula, not how to get from one term to the next. Always find the position-to-term rule using $a_n = a_1 + (n-1)d$ .

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?

The term-to-term rule tells you how to get from one term to the next (subtract 2). The position-to-term rule is a formula like $-2n + 10$ that gives you any term directly from its position.

Why is the common difference negative?

The sequence is decreasing (8, 6, 4, 2...), so each term is smaller than the previous one. When you calculate 6 - 8 = -2, that negative sign shows the sequence goes down by 2 each time.

How do I know which formula to use?

For arithmetic sequences (constant difference between terms), always use $a_n = a_1 + (n-1)d$ . Here, $a_1 = 8$ and $d = -2$ .

Can I check my answer with any term?

Yes! Try $n = 3$ : $-2(3) + 10 = -6 + 10 = 4$ . The 3rd term is indeed 4, so our formula $-2n + 10$ is correct!

What if the sequence continued past 2?

Using our formula: the 5th term would be $-2(5) + 10 = 0$ , the 6th term would be $-2(6) + 10 = -2$ , and so on. The pattern continues infinitely!

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