Determine the Next Number in the Arithmetic Sequence: -6, -3, 0, 3, 6, ?

Q: How do I know this is an arithmetic sequence?

An arithmetic sequence has the same difference between every pair of consecutive terms. Check: $ -3-(-6)=3 $, $ 0-(-3)=3 $, $ 3-0=3 $, $ 6-3=3 $. All differences equal 3!

Q: What if I calculated the differences wrong?

Double-check your subtraction! Remember: when subtracting negative numbers, $ -3-(-6) = -3+6 = 3 $. The pattern should be consistent for all pairs.

Q: Can arithmetic sequences have negative numbers?

Absolutely! This sequence starts with negative numbers and moves to positive. The common difference determines direction: positive difference means increasing, negative means decreasing.

Q: How do I find the 10th term in this sequence?

Use the formula: $ a_n = a_1 + (n-1)d $ where $ a_1 = -6 $ and $ d = 3 $. For the 10th term: $ a_{10} = -6 + (10-1)(3) = -6 + 27 = 21 $

Q: What if the differences aren't the same?

Then it's not an arithmetic sequence! It might be geometric (constant ratio) or follow a different pattern. Always verify the differences are identical before applying arithmetic sequence rules.

Arithmetic Sequences with Common Difference

$-6,-3,0,3,6,\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's complete the next element in our sequence.

00:11 Observe how all the elements are placed on the number line.

00:23 Notice the change between each element on the line.

00:41 We see the sequence is increasing by adding three each time.

00:56 Using this pattern, let's find the next element together.

01:05 And that's how we solve the problem. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$-6,-3,0,3,6,\text{?}$

Step-by-step solution

To solve this problem, we will examine the differences between consecutive terms in the given sequence:
The sequence is $-6, -3, 0, 3, 6$ .

Step 1: Find the difference between consecutive terms.
- From $-6$ to $-3$ : $-3 - (-6) = 3$
- From $-3$ to $0$ : $0 - (-3) = 3$
- From $0$ to $3$ : $3 - 0 = 3$
- From $3$ to $6$ : $6 - 3 = 3$
Step 2: Verify that the difference is consistent. The common difference $d$ is $3$ .
Step 3: Use the common difference to find the next term in the sequence:
From $6$ , add the common difference ( $3$ ): $6 + 3 = 9$ .

Therefore, the next number in the sequence is $9$ .

This matches choice 3 in the provided answer choices.

Final Answer

$9$

Key Points to Remember

Essential concepts to master this topic

Pattern: Check differences between consecutive terms are always equal
Technique: Calculate $6 - 3 = 3$ , then add common difference
Check: Verify all differences equal 3: (-6 to -3), (-3 to 0), (0 to 3), (3 to 6) ✓

Common Mistakes

Avoid these frequent errors

Adding the wrong amount to find next term
Don't just add any number like 1 or 2 = wrong sequence! This ignores the consistent pattern. Always find the common difference first by subtracting consecutive terms, then add that exact difference.

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

How do I know this is an arithmetic sequence?

An arithmetic sequence has the same difference between every pair of consecutive terms. Check: $-3-(-6)=3$ , $0-(-3)=3$ , $3-0=3$ , $6-3=3$ . All differences equal 3!

What if I calculated the differences wrong?

Double-check your subtraction! Remember: when subtracting negative numbers, $-3-(-6) = -3+6 = 3$ . The pattern should be consistent for all pairs.

Can arithmetic sequences have negative numbers?

Absolutely! This sequence starts with negative numbers and moves to positive. The common difference determines direction: positive difference means increasing, negative means decreasing.

How do I find the 10th term in this sequence?

Use the formula: $a_n = a_1 + (n-1)d$ where $a_1 = -6$ and $d = 3$ . For the 10th term: $a_{10} = -6 + (10-1)(3) = -6 + 27 = 21$

What if the differences aren't the same?

Then it's not an arithmetic sequence! It might be geometric (constant ratio) or follow a different pattern. Always verify the differences are identical before applying arithmetic sequence rules.

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