Determine the Term-to-Term Rule for the Sequence: 33, 25, 17, 9

Q: Why is the common difference negative?

The common difference is negative because each term is smaller than the previous one. When you go from 33 to 25, you subtract 8, so d = -8.

Q: What's the difference between -8n + 41 and 41 - 8n?

They're exactly the same! Just written in different order. Both mean 'start with 41 and subtract 8n'. Choose whichever format matches your answer choices.

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

Yes! Just substitute the position number for n. For example, the 10th term would be $ -8(10) + 41 = -80 + 41 = -39 $.

Q: What if I get confused about the signs?

Remember: if the sequence goes down, the coefficient of n is negative. If it goes up, the coefficient is positive. The sequence 33, 25, 17, 9 clearly goes down!

Arithmetic Sequences with Negative Common Differences

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

33, 25, 17, 9, ...

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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

00:14 Notice the constant difference between terms

00:20 This is the constant difference

00:26 Use the formula to describe an arithmetic sequence

00:34 Substitute appropriate values and solve to find the sequence formula

00:46 Expand brackets properly, multiply by each factor

00:54 Continue solving

01:04 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?

33, 25, 17, 9, ...

Step-by-step solution

To solve this problem, we'll analyze the sequence to find the common difference, determine the general form of the sequence formula, and then verify which of the given multiple-choice options corresponds to our result.

Step 1: Calculate the common difference $d$ .
The sequence is: 33, 25, 17, 9,...
Calculate differences between terms:
$25 - 33 = -8$
$17 - 25 = -8$
$9 - 17 = -8$
The common difference $d$ is $-8$ .
Step 2: Identify the first term $a_1$ .
The first term of the sequence is 33.
Step 3: Write the formula for the $n$ -th term of the sequence.
Using the formula $a_n = a_1 + (n-1)d$ , substitute $a_1 = 33$ and $d = -8$ :
$a_n = 33 + (n-1)(-8)$
Simplifying, we have:
$a_n = 33 - 8n + 8$
$a_n = 41 - 8n$
Step 4: Compare the formula $a_n = 41 - 8n$ with the answer choices to find the matching option.

The term-to-term rule for this sequence is $-8n + 41$ , which corresponds to choice 2.

Final Answer

$-8n+41$

Key Points to Remember

Essential concepts to master this topic

Common Difference: Subtract consecutive terms to find the pattern
Formula: Use $a_n = a_1 + (n-1)d$ where d = -8
Check: Verify: $-8(1) + 41 = 33$ , $-8(2) + 41 = 25$ ✓

Common Mistakes

Avoid these frequent errors

Writing the formula as positive slope instead of negative
Don't write 8n + 41 when the sequence decreases = wrong answers for all terms! The sequence goes down by 8 each time, so the coefficient must be negative. Always check if terms increase (+) or decrease (-) to determine the sign.

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 is the common difference negative?

The common difference is negative because each term is smaller than the previous one. When you go from 33 to 25, you subtract 8, so d = -8.

How do I know which answer choice is correct?

Test each formula with n = 1. The correct formula should give you 33 (the first term). Only $-8n + 41$ gives $-8(1) + 41 = 33$ .

What's the difference between -8n + 41 and 41 - 8n?

They're exactly the same! Just written in different order. Both mean 'start with 41 and subtract 8n'. Choose whichever format matches your answer choices.

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

Yes! Just substitute the position number for n. For example, the 10th term would be $-8(10) + 41 = -80 + 41 = -39$ .

What if I get confused about the signs?

Remember: if the sequence goes down, the coefficient of n is negative. If it goes up, the coefficient is positive. The sequence 33, 25, 17, 9 clearly goes down!

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