Find the Term-to-Term Rule: Analyzing the Sequence 10, 6, 2, -2

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

The term-to-term rule describes how to get from one term to the next (like 'subtract 4'). The nth term formula lets you find any term directly using its position number, like $ T_n = 14 - 4n $.

Q: Why is the common difference negative?

Because each term is smaller than the previous one! When 6 - 10 = -4, that negative sign tells us the sequence is decreasing by 4 each time.

Q: How do I remember the nth term formula?

Think: Start + (steps × difference). So T_n = first term + (n-1) × common difference. The (n-1) counts how many steps from the first term to the nth term.

Q: What if I get the wrong sign in my answer?

Double-check your common difference calculation! Remember: later term - earlier term. If the sequence decreases, your difference should be negative.

Arithmetic Sequences with Negative Differences

What is the term-to-term rule for the sequence below?

$10,6,2,-2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the sequence formula

00:05 Identify the first term according to the given data

00:10 Note the constant difference between terms

00:19 This is the constant difference

00:24 Use the formula for describing an arithmetic sequence

00:31 Substitute appropriate values and solve to find the sequence formula

00:45 Properly open parentheses, multiply by each factor

00:53 Continue solving

01:01 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 for the sequence below?

$10,6,2,-2$

Step-by-step solution

To determine the term-to-term rule for the sequence $10, 6, 2, -2$ , we need to find the pattern in changes between terms:

Calculate the differences: $6 - 10 = -4$ , $2 - 6 = -4$ , $-2 - 2 = -4$ .
These consistent differences ( $-4$ ) indicate an arithmetic sequence with a common difference of $-4$ .

The general term of an arithmetic sequence is given by: $T_n = a + (n-1)d$ , where $a$ is the first term and $d$ is the common difference.

Substitute $a = 10$ and $d = -4$ :

$T_n = 10 + (n-1)(-4)$

$T_n = 10 - 4(n-1)$

$T_n = 10 - 4n + 4$

Simplifying gives: $T_n = 14 - 4n$

Thus, the term-to-term rule for the sequence is $14 - 4n$ .

Final Answer

$14-4n$

Key Points to Remember

Essential concepts to master this topic

Pattern: Find common difference by subtracting consecutive terms consistently
Formula: Use T_n = a + (n-1)d where a=10, d=-4
Verify: Check T_1 = 14-4(1) = 10, T_2 = 14-4(2) = 6 ✓

Common Mistakes

Avoid these frequent errors

Confusing term-to-term rule with nth term formula
Don't just say 'subtract 4 each time' when asked for the rule = incomplete answer! The question wants the algebraic formula to find any term directly. Always express as T_n = 14 - 4n to show the complete relationship.

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?

The term-to-term rule describes how to get from one term to the next (like 'subtract 4'). The nth term formula lets you find any term directly using its position number, like $T_n = 14 - 4n$ .

Why is the common difference negative?

Because each term is smaller than the previous one! When 6 - 10 = -4, that negative sign tells us the sequence is decreasing by 4 each time.

How do I remember the nth term formula?

Think: Start + (steps × difference). So T_n = first term + (n-1) × common difference. The (n-1) counts how many steps from the first term to the nth term.

Can I check my formula with any term?

Yes! Pick any term from the sequence and substitute its position into your formula. For example: $T_3 = 14 - 4(3) = 2$ ✓ matches the third term.

What if I get the wrong sign in my answer?

Double-check your common difference calculation! Remember: later term - earlier term. If the sequence decreases, your difference should be negative.

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