Finding The 7th Term Of An Arithmetic Sequence
Hey guys! Today, we're diving into the world of arithmetic sequences and tackling a common question: How do we find a specific term in a sequence when we have an explicit formula? Specifically, we'll be figuring out what is the seventh term in the arithmetic sequence defined by the explicit formula aₙ = 5n + 3. Don't worry; it's way simpler than it sounds! So, let's get started and break this down step by step.
Understanding Arithmetic Sequences and Explicit Formulas
Before we jump into the calculation, let's make sure we're all on the same page about what arithmetic sequences and explicit formulas are. An arithmetic sequence is basically a list of numbers where the difference between any two consecutive terms is always the same. This consistent difference is called the common difference. For example, in the sequence 2, 4, 6, 8, 10, the common difference is 2 because you add 2 to each term to get the next one. Now, an explicit formula is a handy tool that allows us to directly calculate any term in the sequence without having to know the previous terms. It's like having a secret code that instantly reveals the value of any term you want. The general form of an explicit formula for an arithmetic sequence is aₙ = a₁ + (n - 1)d, where aₙ is the nth term, a₁ is the first term, n is the term number, and d is the common difference. However, in our case, we're given a slightly different explicit formula, aₙ = 5n + 3, which is already tailored to this specific sequence.
Applying the Explicit Formula
Now comes the fun part: using the explicit formula to find the seventh term. The formula we have is aₙ = 5n + 3. We want to find the seventh term, which means we're looking for a₇. To do this, we simply substitute n with 7 in the formula. So, a₇ = 5(7) + 3. Following the order of operations, we first multiply 5 by 7, which gives us 35. Then, we add 3 to 35, resulting in 38. Therefore, a₇ = 38. That's it! The seventh term in the arithmetic sequence defined by the explicit formula aₙ = 5n + 3 is 38. See? It wasn't so bad after all!
Step-by-Step Calculation
Let's quickly recap the step-by-step calculation to solidify our understanding:
- Identify the explicit formula: aₙ = 5n + 3
- Determine the term number: We want to find the seventh term, so n = 7
- Substitute n with 7 in the formula: a₇ = 5(7) + 3
- Perform the multiplication: 5(7) = 35
- Add the remaining constant: 35 + 3 = 38
- State the result: a₇ = 38
By following these simple steps, you can easily find any term in an arithmetic sequence when you have the explicit formula. Remember, the key is to substitute the term number into the formula and then simplify.
Alternative Methods (If Applicable)
While using the explicit formula is the most direct way to find the seventh term, let's briefly consider alternative methods, even though they might be less efficient in this case. One approach could be to find the first few terms of the sequence and then identify the pattern to reach the seventh term. To find the first term (a₁), we substitute n with 1 in the formula: a₁ = 5(1) + 3 = 8. Similarly, to find the second term (a₂), we substitute n with 2: a₂ = 5(2) + 3 = 13. We can continue this process to find more terms: a₃ = 18, a₄ = 23, a₅ = 28, a₆ = 33, and finally, a₇ = 38. As you can see, this method works, but it requires more calculations and can be time-consuming, especially if you need to find a term that is further down the sequence. Another method involves recognizing that the coefficient of n in the explicit formula (which is 5 in our case) represents the common difference of the arithmetic sequence. Knowing this, you could start with the first term (8) and repeatedly add the common difference (5) until you reach the seventh term. However, this method is also less efficient than directly using the explicit formula.
Real-World Applications
Arithmetic sequences might seem like an abstract mathematical concept, but they actually have numerous applications in the real world. For example, they can be used to model situations involving linear growth or decay. Imagine you're saving money, and you deposit a fixed amount each month. The total amount of money you have saved over time would form an arithmetic sequence. Similarly, if you're paying off a loan with fixed monthly payments, the remaining balance on the loan would also follow an arithmetic sequence. Arithmetic sequences are also used in various fields such as computer science, finance, and physics. In computer science, they can be used to analyze the performance of algorithms. In finance, they can be used to calculate compound interest. And in physics, they can be used to model the motion of objects with constant acceleration. Understanding arithmetic sequences and how to work with them can provide valuable insights into many real-world phenomena.
Practice Problems
To further solidify your understanding of arithmetic sequences and explicit formulas, here are a few practice problems for you to try:
- Find the tenth term in the arithmetic sequence defined by the explicit formula aₙ = 3n - 2.
- What is the fifth term in the arithmetic sequence defined by the explicit formula aₙ = -2n + 7?
- Determine the fifteenth term in the arithmetic sequence defined by the explicit formula aₙ = n + 4.
Pro Tip: Remember to substitute the term number into the explicit formula and then simplify. And don't be afraid to use a calculator if needed!
Conclusion
So, there you have it! Finding the seventh term in an arithmetic sequence using an explicit formula is a straightforward process. By understanding the concept of arithmetic sequences and how to apply the explicit formula, you can easily calculate any term in the sequence. Remember to identify the explicit formula, substitute the term number, and simplify. With a little practice, you'll become a pro at working with arithmetic sequences. Keep practicing, and you'll be amazed at how useful these concepts can be in various real-world scenarios. Happy calculating!
