Number Pattern: Finding The 7th To 10th Terms

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Let's dive into this number pattern and figure out what the 7th, 8th, 9th, and 10th terms are! We've got a sequence that starts with sa 774, 729, 714, 699, 684, and 669. Our mission is to continue this pattern and identify the next four terms. So, grab your thinking caps, and let's get started!

Analyzing the Number Pattern

Alright, first things first, we need to figure out what's going on in this sequence. Number patterns are all about finding the hidden rule that connects each number to the next. So, let's take a close look at the differences between consecutive terms.

  • From 774 to 729: The difference is 774 - 729 = 45
  • From 729 to 714: The difference is 729 - 714 = 15
  • From 714 to 699: The difference is 714 - 699 = 15
  • From 699 to 684: The difference is 699 - 684 = 15
  • From 684 to 669: The difference is 684 - 669 = 15

Okay, it looks like after the initial drop of 45, the sequence decreases by 15 each time. This makes it an arithmetic sequence, which is just a fancy way of saying that we're subtracting the same number over and over again.

Finding the Next Terms

Now that we know the rule, finding the next terms is a piece of cake! We just keep subtracting 15 from the last known term.

  1. 6th term: 669
  2. 7th term: 669 - 15 = 654
  3. 8th term: 654 - 15 = 639
  4. 9th term: 639 - 15 = 624
  5. 10th term: 624 - 15 = 609

So, the 7th, 8th, 9th, and 10th terms in the sequence are 654, 639, 624, and 609, respectively.

The Answer

  • 7th term: 654
  • 8th term: 639
  • 9th term: 624
  • 10th term: 609

In conclusion, by analyzing the pattern and identifying the constant difference, we successfully found the 7th, 8th, 9th, and 10th terms of the given sequence. This is a classic example of how recognizing patterns can help us solve mathematical problems. Great job, guys! Let's tackle more number patterns next time!

Understanding Arithmetic Sequences

To truly master these types of problems, let's explore arithmetic sequences a bit more. Arithmetic sequences are sequences where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted as 'd'.

The general form of an arithmetic sequence is:

a, a + d, a + 2d, a + 3d, ...

Where:

  • 'a' is the first term
  • 'd' is the common difference

In our example:

  • The sequence (after the first term sa) is 774, 729, 714, 699, 684, 669, ...
  • The common difference 'd' is -15 (since we are subtracting 15 each time)

Formula for the nth Term

We can also use a formula to find any term in the arithmetic sequence directly. The formula for the nth term (an) is:

an = a + (n - 1)d

Where:

  • an is the nth term
  • a is the first term
  • n is the term number
  • d is the common difference

Let's verify this formula by finding the 7th term in our sequence:

a = 774 n = 7 d = -15

a7 = 774 + (7 - 1) * (-15) a7 = 774 + (6) * (-15) a7 = 774 - 90 a7 = 684

Wait a minute! We got 684, but we previously found the 7th term to be 654. What went wrong?

Ah, it seems we made a mistake in considering 774 as the first term for calculating subsequent terms using the arithmetic sequence formula. The common difference of -15 applies from the second term onward (729). So, let’s correct our approach.

To correctly use the formula, we should consider 729 as our 'a' (the first term we're using for our calculations) and adjust the 'n' value accordingly. If we want to find the 7th term of the original sequence, that’s the 6th term after 729. So, for our formula, n = 6.

a = 729 n = 6 (since we want the 6th term after 729 to correspond to the 7th term in the original sequence) d = -15

an = a + (n - 1)d an = 729 + (6 - 1) * (-15) an = 729 + (5) * (-15) an = 729 - 75 an = 654

Okay, that's more like it! The 7th term is indeed 654, which matches our earlier calculation. This formula is super handy for finding any term in the sequence without having to list all the terms before it.

Real-World Applications

You might be wondering, where do we actually use arithmetic sequences in real life? Well, they pop up in various scenarios:

  • Simple Interest: The amount of interest earned each year on a fixed deposit forms an arithmetic sequence.
  • Salary Increments: If you get a fixed salary increase each year, your salary over the years forms an arithmetic sequence.
  • Stacking Objects: If you're stacking objects like cans in a pyramid shape where each layer has one less can, the number of cans in each layer forms an arithmetic sequence.
  • Depreciation: The value of an asset that decreases by a fixed amount each year follows an arithmetic sequence.

Understanding arithmetic sequences can help you model and predict these real-world scenarios.

Practice Problems

Want to test your understanding? Try these practice problems:

  1. Find the 15th term of the arithmetic sequence: 3, 7, 11, 15, ...
  2. The first term of an arithmetic sequence is 5, and the common difference is -2. Find the first 5 terms of the sequence.
  3. An arithmetic sequence has the first term as 10 and the 10th term as 55. Find the common difference.

Solving these problems will solidify your understanding of arithmetic sequences and make you a pattern-finding pro! Keep practicing, and you'll be spotting these sequences everywhere!

Conclusion

So there you have it, we've successfully navigated through a number pattern, identified the underlying arithmetic sequence, and found the 7th, 8th, 9th and 10th terms. We also delved deeper into arithmetic sequences, explored the formula for finding the nth term, and looked at some real-world applications. Keep honing your pattern-recognition skills, and you'll be amazed at how these mathematical concepts can help you understand the world around you.

Remember, math isn't just about numbers and formulas; it's about problem-solving, critical thinking, and seeing the hidden connections in everything. Keep exploring, keep questioning, and keep learning! You've got this!