Complete the Sequence: Finding Numbers After 6,500, 6,400, 6,300

Q: What if I accidentally use positive 100 instead of negative 100?

You'll get the wrong direction! Using +100 would give you 6,400, 6,500, 6,600 - which goes backwards from the given pattern. Always check your difference sign carefully.

Q: Do I always subtract 100 from each term?

For this specific sequence, yes! But remember: the common difference depends on the pattern. Other sequences might decrease by 50, 25, or any other amount.

Q: What if the numbers don't have a constant difference?

Then it's not an arithmetic sequence! Look for other patterns like multiplication, squares, or more complex rules. But this problem clearly shows a constant decrease of 100.

Arithmetic Sequences with Decreasing Values

Complete the sequence:

$6{,}500,\ 6{,}400,\ 6{,}300, \ \ldots$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the sequence:

$6{,}500,\ 6{,}400,\ 6{,}300, \ \ldots$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the given numbers and the rule or pattern.
Step 2: Determine the common difference, which is $-100$ .
Step 3: Use this pattern to continue the sequence.

Now, let's work through each step:
Step 1: The initial terms of the sequence are $6500, 6400, 6300$ .
Step 2: Observing the sequence, each number decreases by 100 from the previous number. This indicates a common difference of $-100$ .
Step 3: Continuing this pattern:
If the last given number is 6300, the next number would be $6300 - 100 = 6200$ .
Similarly, we continue:
$6200 - 100 = 6100$
$6100 - 100 = 6000$

Therefore, continuing the sequence results in the terms: $6200, 6100, 6000$ .

Therefore, the solution to the problem is $6200, 6100, 6000$ .

Final Answer

$6{,}200,\ 6{,}100,\ 6{,}000$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Each term decreases by constant difference
Technique: Common difference = 6,400 - 6,500 = -100
Check: Verify pattern holds: 6,300 - 100 = 6,200 ✓

Common Mistakes

Avoid these frequent errors

Finding difference in wrong direction
Don't calculate 6,500 - 6,400 = +100 and add to continue the sequence = wrong direction! This gives increasing values when the pattern is clearly decreasing. Always calculate second term minus first term to get the correct sign.

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 the sequence is increasing or decreasing?

Look at the direction from one term to the next! If the numbers get smaller (like 6,500 → 6,400 → 6,300), it's decreasing. If they get larger, it's increasing.

What if I accidentally use positive 100 instead of negative 100?

You'll get the wrong direction! Using +100 would give you 6,400, 6,500, 6,600 - which goes backwards from the given pattern. Always check your difference sign carefully.

Do I always subtract 100 from each term?

For this specific sequence, yes! But remember: the common difference depends on the pattern. Other sequences might decrease by 50, 25, or any other amount.

How can I double-check my pattern is correct?

Test it on the given terms! If your pattern is right, applying it should recreate the sequence: $6,500 - 100 = 6,400$ and $6,400 - 100 = 6,300$ ✓

What if the numbers don't have a constant difference?

Then it's not an arithmetic sequence! Look for other patterns like multiplication, squares, or more complex rules. But this problem clearly shows a constant decrease of 100.

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