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

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 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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