Find the Term-to-Term Rule in the Sequence: 40, 35, 30

Q: What's the difference between term-to-term rule and the formula?

The term-to-term rule tells you how to get from one term to the next (subtract 5). The position formula $ 45-5n $ lets you find any term directly using its position number.

Q: Why is it 45-5n and not 40-5n?

When n=1, we need the first term (40). Using $ 45-5(1) = 40 $ ✓. But $ 40-5(1) = 35 $ gives the wrong first term!

Q: How do I know if the sequence is decreasing?

Look at the common difference! If it's negative (like -5), the sequence decreases. If it's positive, the sequence increases.

Q: What if I can't see the pattern right away?

Always calculate the differences step by step: second term - first term, then third term - second term. If they're the same, you have an arithmetic sequence!

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

Yes! For any arithmetic sequence, the formula is $ a + (n-1)d $ where a is the first term and d is the common difference.

Arithmetic Sequences with Decreasing Terms

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

40 ,35, 30

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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

00:12 Note the constant difference between terms

00:17 This is the constant difference

00:23 Use the formula to describe an arithmetic sequence

00:32 Substitute appropriate values and solve to find the sequence formula

00:50 Properly open parentheses, multiply by each factor

00:56 Continue solving

01:02 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?

40 ,35, 30

Step-by-step solution

To solve this problem, we'll identify the term-to-term rule for the given sequence: 40, 35, 30.

Step 1: Calculate the difference between consecutive terms.
The second term (35) is 5 less than the first term (40): $35 - 40 = -5$ .
The third term (30) is 5 less than the second term (35): $30 - 35 = -5$ .
The difference between consecutive terms is consistent at $-5$ .

Step 2: Express the term-to-term rule.
Since each term is obtained by subtracting 5 from the previous term, the term-to-term rule is:
"Subtract 5 from the previous term to get the next term."

Step 3: Compare to given answer choices.
The correct expression is consistent with choice 1: $45-5n$ . This expression generates the sequence by considering $n$ starting from 1.

Therefore, the term-to-term rule for the sequence is correctly represented by the formula $45-5n$ .

Final Answer

$45-5n$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Find the common difference between consecutive terms first
Technique: Calculate 35 - 40 = -5, then 30 - 35 = -5
Verification: Check formula works: $45-5(1) = 40$ , $45-5(2) = 35$ ✓

Common Mistakes

Avoid these frequent errors

Confusing term-to-term rule with position-to-term formula
Don't think the term-to-term rule is just 'subtract 5' without checking the position formula! This misses how the sequence relates to term positions. Always verify your formula generates the correct sequence when n = 1, 2, 3.

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 rule and the formula?

The term-to-term rule tells you how to get from one term to the next (subtract 5). The position formula $45-5n$ lets you find any term directly using its position number.

Why is it 45-5n and not 40-5n?

When n=1, we need the first term (40). Using $45-5(1) = 40$ ✓. But $40-5(1) = 35$ gives the wrong first term!

How do I know if the sequence is decreasing?

Look at the common difference! If it's negative (like -5), the sequence decreases. If it's positive, the sequence increases.

What if I can't see the pattern right away?

Always calculate the differences step by step: second term - first term, then third term - second term. If they're the same, you have an arithmetic sequence!

Can I use this method for any arithmetic sequence?

Yes! For any arithmetic sequence, the formula is $a + (n-1)d$ where a is the first term and d is the common difference.

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