Complete the Sequence: Finding the Pattern in 1,000, 2,000, 3,000,...

Q: How do I know if a sequence is arithmetic?

Check if the difference between consecutive terms is always the same. In this case: 2000 - 1000 = 1000 and 3000 - 2000 = 1000, so it's arithmetic!

Q: Can arithmetic sequences go backwards?

Yes! If the common difference is negative, the sequence decreases. For example: 10, 7, 4, 1... has a common difference of -3.

Q: How many terms should I find?

The question will tell you! Here we needed the next three terms after 3000, which are $ 4000, 5000, 6000 $.

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

Start by calculating differences between consecutive terms. Write them down: 2000 - 1000 = ?, 3000 - 2000 = ?. If they're equal, you found your pattern!

Arithmetic Sequences with Constant Differences

Complete the sequence:

$1{,}000,\ 2{,}000,\ 3{,}000, \ \ldots$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the sequence:

$1{,}000,\ 2{,}000,\ 3{,}000, \ \ldots$

Step-by-step solution

To solve this problem, let's determine the pattern in the given sequence:

Given Sequence: $1000, 2000, 3000, \ldots$
Step 1: Calculate the common difference.

The difference between 2000 and 1000 is $2000 - 1000 = 1000$ .
The difference between 3000 and 2000 is $3000 - 2000 = 1000$ .

Step 2: Use the common difference to find the subsequent terms.

The next term after 3000 is $3000 + 1000 = 4000$ .
Following that, the next term is $4000 + 1000 = 5000$ .
And then $5000 + 1000 = 6000$ .

Step 3: Compare with answer choices.
Choice 4: $4000, 5000, 6000$ matches the calculated sequence.

Therefore, the completed sequence is $4000, 5000, 6000$ .

Final Answer

$4{,}000,\ 5{,}000,\ 6{,}000$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Find the common difference between consecutive terms
Technique: Add the difference repeatedly: 3000 + 1000 = 4000
Check: Verify each term differs by the same amount ✓

Common Mistakes

Avoid these frequent errors

Assuming the pattern changes or becomes more complex
Don't look for complicated patterns when a simple one exists = wrong continuation! Students often overthink and create complex rules when the sequence just adds 1000 each time. Always identify the simplest consistent pattern first.

Practice Quiz

Test your knowledge with interactive questions

Complete the sequence:

$1{,}113,\ 1{,}112,\ 1{,}111, \ \ldots$

FAQ

Everything you need to know about this question

How do I know if a sequence is arithmetic?

Check if the difference between consecutive terms is always the same. In this case: 2000 - 1000 = 1000 and 3000 - 2000 = 1000, so it's arithmetic!

What if the numbers get really big?

The pattern stays the same! Whether you're adding 1, 10, or 1000, just keep adding the same amount to get the next term. Size doesn't change the rule.

Can arithmetic sequences go backwards?

Yes! If the common difference is negative, the sequence decreases. For example: 10, 7, 4, 1... has a common difference of -3.

How many terms should I find?

The question will tell you! Here we needed the next three terms after 3000, which are $4000, 5000, 6000$ .

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

Start by calculating differences between consecutive terms. Write them down: 2000 - 1000 = ?, 3000 - 2000 = ?. If they're equal, you found your pattern!

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