Determine the Term-to-Term Rule for the Sequence: 13, 10, 7, 4

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

Great question! The term-to-term rule can mean the pattern (subtract 3) OR the formula. Here, they want the algebraic formula $ 16 - 3n $ that gives any term directly.

Q: Why is the common difference negative?

Because the sequence is decreasing! We go from 13 to 10 to 7 to 4, so each term is 3 less than the previous one. That's why d = -3.

Q: How do I remember the arithmetic sequence formula?

Think: Start + Steps × Step Size. You start at $ a_1 $, take $ (n-1) $ steps, each step is size $ d $!

Q: What if I get a positive common difference?

Then you made an error! This sequence decreases, so the common difference must be negative. Always subtract: later term - earlier term.

Arithmetic Sequences with Negative Common Differences

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

13, 10, 7, 4, ...

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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 Notice the constant difference between terms

00:19 This is the constant difference

00:26 Use the formula to describe an arithmetic sequence

00:33 Substitute appropriate values and solve to find the sequence formula

00:50 Properly expand brackets, multiply by each factor

01:01 Continue solving

01:08 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?

13, 10, 7, 4, ...

Step-by-step solution

To identify the term-to-term rule of the sequence $13, 10, 7, 4, \ldots$ , let's analyze the differences between consecutive terms:

From 13 to 10: $10 - 13 = -3$
From 10 to 7: $7 - 10 = -3$
From 7 to 4: $4 - 7 = -3$

The common difference is $-3$ , confirming the sequence is arithmetic. In an arithmetic sequence, we use the formula $a_n = a_1 + (n-1) \times d$ .

Here, $a_1 = 13$ , and the common difference $d = -3$ . Plug these into the formula:

$a_n = 13 + (n-1)(-3)$

Simplify the expression:

$a_n = 13 - 3(n-1)$

$a_n = 13 - 3n + 3$

$a_n = 16 - 3n$

Thus, the term-to-term rule of the sequence is $16-3n$ . This matches choice 3 from the provided options.

Final Answer

$16-3n$

Key Points to Remember

Essential concepts to master this topic

Pattern: Find common difference by subtracting consecutive terms
Formula: Use $a_n = a_1 + (n-1)d$ where d = -3
Check: Test formula with known terms: $16 - 3(1) = 13$ ✓

Common Mistakes

Avoid these frequent errors

Confusing term-to-term rule with nth term formula
Don't think the term-to-term rule means 'subtract 3 each time' = missing the actual formula! The question asks for the algebraic expression. Always use $a_n = a_1 + (n-1)d$ and simplify to get the nth term formula.

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

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

Great question! The term-to-term rule can mean the pattern (subtract 3) OR the formula. Here, they want the algebraic formula $16 - 3n$ that gives any term directly.

Why is the common difference negative?

Because the sequence is decreasing! We go from 13 to 10 to 7 to 4, so each term is 3 less than the previous one. That's why d = -3.

How do I remember the arithmetic sequence formula?

Think: Start + Steps × Step Size. You start at $a_1$ , take $(n-1)$ steps, each step is size $d$ !

Can I check my formula with any term?

Yes! Pick any position you know. For example, the 2nd term should be 10: $16 - 3(2) = 16 - 6 = 10$ ✓

What if I get a positive common difference?

Then you made an error! This sequence decreases, so the common difference must be negative. Always subtract: later term - earlier term.

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