Sequence Challenge: Identify the Next Number in 653,150, 653,250, ...

Q: How do I find the common difference?

Subtract the first term from the second term: $ 653{,}250 - 653{,}150 = 100 $. This common difference tells you how much to add each time!

Q: What if the numbers are decreasing instead of increasing?

The common difference would be negative! For example, if you had 500, 400, 300, the common difference is -100, so you'd subtract 100 each time.

Q: Can I skip ahead to find the 10th term without writing all terms?

Yes! Use the formula: First term + (position - 1) × common difference. For the 10th term: $ 653{,}150 + (10-1) \times 100 = 654{,}050 $

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

Check that the difference between consecutive terms is always the same. If it changes, it's not arithmetic. In our case, every step increases by exactly 100.

Q: What if I made an error in my calculation?

Always double-check by working backwards! Take your answer and subtract the common difference - you should get the previous term in the sequence.

Arithmetic Sequences with Constant Differences

Complete the sequence:

$653{,}150,\ 653{,}250, \ \ldots$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the sequence:

$653{,}150,\ 653{,}250, \ \ldots$

Step-by-step solution

To solve the problem, we identify the sequence pattern:

Determine the difference between the given terms: $653,250 - 653,150 = 100$ .
Since the difference between consecutive terms is 100, this indicates the rule of the sequence.
To find the next terms, continue adding 100 to the last known term:

Next term after 653,250: $653,250 + 100 = 653,350$ .
Next term after 653,350: $653,350 + 100 = 653,450$ .
Next term after 653,450: $653,450 + 100 = 653,550$ .

Thus, the completed sequence is $653,150, 653,250, 653,350, 653,450, 653,550$ .

Therefore, the next three numbers in the sequence are $653{,}350, 653{,}450, 653{,}550$ .

Final Answer

$653{,}350,\ 653{,}450,\ 653{,}550$

Key Points to Remember

Essential concepts to master this topic

Pattern Rule: Find the common difference between consecutive terms
Technique: Add the common difference: $653{,}250 + 100 = 653{,}350$
Check: Verify each term increases by exactly 100: $653{,}450 - 653{,}350 = 100$ ✓

Common Mistakes

Avoid these frequent errors

Adding different amounts instead of the consistent common difference
Don't add random numbers like 1, 10, or 50 = wrong sequence! This breaks the pattern and creates an invalid sequence. Always add the exact same common difference (100 in this case) to each term.

Practice Quiz

Test your knowledge with interactive questions

Complete the sequence:

$20{,}000,\ 20{,}001,\ 20{,}002, \ \ldots$

FAQ

Everything you need to know about this question

How do I find the common difference?

Subtract the first term from the second term: $653{,}250 - 653{,}150 = 100$ . This common difference tells you how much to add each time!

What if the numbers are decreasing instead of increasing?

The common difference would be negative! For example, if you had 500, 400, 300, the common difference is -100, so you'd subtract 100 each time.

Can I skip ahead to find the 10th term without writing all terms?

Yes! Use the formula: First term + (position - 1) × common difference. For the 10th term: $653{,}150 + (10-1) \times 100 = 654{,}050$

How do I know this is definitely an arithmetic sequence?

Check that the difference between consecutive terms is always the same. If it changes, it's not arithmetic. In our case, every step increases by exactly 100.

What if I made an error in my calculation?

Always double-check by working backwards! Take your answer and subtract the common difference - you should get the previous term in the sequence.

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