Complete the Sequence: An Arithmetic Progression with Million Multiples

Q: How do I know this is an arithmetic sequence and not geometric?

In an arithmetic sequence, you add the same number each time. In a geometric sequence, you multiply by the same number. Since $ 2{,}000{,}000 - 1{,}000{,}000 = 1{,}000{,}000 $, we're adding!

Q: What if the numbers were much smaller, like 2, 4, 6?

The same method works! Find the difference: $ 4 - 2 = 2 $, then add 2 to get the next terms: 8, 10, 12. The size of numbers doesn't change the pattern.

Q: Can I write a formula for the nth term?

Yes! For this sequence, the formula is $ a_n = 1{,}000{,}000 \times n $. The general formula for arithmetic sequences is $ a_n = a_1 + (n-1)d $ where d is the common difference.

Q: Why do all these numbers end in so many zeros?

These are multiples of one million! When you multiply by powers of 10 (like 1,000,000), you get lots of zeros. This makes the arithmetic easier since you're just working with 1, 2, 3, 4, 5 and adding zeros.

Arithmetic Sequences with Million Intervals

Complete the sequence:

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

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the sequence:

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

Step-by-step solution

To solve this problem, we will follow these steps:

Step 1: Identify the pattern by examining the given numbers in the sequence.
Step 2: Calculate the difference between the first two numbers in the sequence.
Step 3: Use this difference to find the next numbers in the sequence.

Now, let's work through each step:

Step 1: The given sequence starts with the numbers $1,000,000$ and $2,000,000$ .

Step 2: We calculate the difference between these numbers: $2,000,000 - 1,000,000 = 1,000,000$ This suggests that each term in the sequence increases by $1,000,000$ .

Step 3: Using this pattern, we add $1,000,000$ to the last known term to find the next terms: $2,000,000 + 1,000,000 = 3,000,000$ $3,000,000 + 1,000,000 = 4,000,000$ $4,000,000 + 1,000,000 = 5,000,000$ Thus, the next terms in the sequence are $3,000,000$ , $4,000,000$ , and $5,000,000$ .

Therefore, the solution to the problem is:
$3{,}000{,}000,\ 4{,}000{,}000,\ 5{,}000{,}000$ .

Final Answer

$3{,}000{,}000,\ 4{,}000{,}000, \ 5{,}000{,}000$

Key Points to Remember

Essential concepts to master this topic

Pattern: Find the common difference between consecutive terms
Technique: Add $1{,}000{,}000$ repeatedly: $2{,}000{,}000 + 1{,}000{,}000 = 3{,}000{,}000$
Check: Verify each term increases by the same amount: $4{,}000{,}000 - 3{,}000{,}000 = 1{,}000{,}000$ ✓

Common Mistakes

Avoid these frequent errors

Assuming the pattern changes or becomes more complex
Don't look for complicated patterns like adding different amounts each time = wrong sequence! Students often overthink simple arithmetic sequences. Always check if the difference between consecutive terms stays constant throughout.

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 know this is an arithmetic sequence and not geometric?

In an arithmetic sequence, you add the same number each time. In a geometric sequence, you multiply by the same number. Since $2{,}000{,}000 - 1{,}000{,}000 = 1{,}000{,}000$ , we're adding!

What if the numbers were much smaller, like 2, 4, 6?

The same method works! Find the difference: $4 - 2 = 2$ , then add 2 to get the next terms: 8, 10, 12. The size of numbers doesn't change the pattern.

Can I write a formula for the nth term?

Yes! For this sequence, the formula is $a_n = 1{,}000{,}000 \times n$ . The general formula for arithmetic sequences is $a_n = a_1 + (n-1)d$ where d is the common difference.

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

Start simple! Calculate the difference between the first two terms. If that same difference works for other consecutive pairs, you've found your pattern. Don't worry about complex formulas until you understand the basic pattern.

Why do all these numbers end in so many zeros?

These are multiples of one million! When you multiply by powers of 10 (like 1,000,000), you get lots of zeros. This makes the arithmetic easier since you're just working with 1, 2, 3, 4, 5 and adding zeros.

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