Find The 10th Term Of An Arithmetic Sequence
Hey guys! Today, we're diving into the fascinating world of arithmetic sequences. Imagine a line of numbers where the difference between each consecutive number is always the same. That's an arithmetic sequence! Think of it like climbing stairs where each step is the same height. We're going to tackle a fun problem: given the 5th term and the 9th term of an arithmetic sequence, how do we find the 10th term? Let's break it down step-by-step.
What is an Arithmetic Sequence?
Before we jump into the problem, let's make sure we're all on the same page about what an arithmetic sequence actually is. At its heart, an arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted by 'd'.
For example, the sequence 2, 5, 8, 11, 14... is an arithmetic sequence because we're adding 3 each time (the common difference is 3). Similarly, 10, 7, 4, 1, -2... is also an arithmetic sequence, but this time we're subtracting 3 each time (the common difference is -3).
The general form of an arithmetic sequence can be written as:
a, a + d, a + 2d, a + 3d, ...
Where:
- 'a' is the first term of the sequence.
- 'd' is the common difference.
The nth term of an arithmetic sequence, often denoted as an, can be found using the following formula:
an = a + (n - 1)d
This formula is super important because it allows us to find any term in the sequence without having to list out all the terms before it. Now that we have a solid understanding of arithmetic sequences, let's get back to our problem!
Problem: Finding the 10th Term
Okay, so here's the problem we're going to solve:
We have an arithmetic sequence where the 5th term is 8 and the 9th term is 20. What is the 10th term?
This might seem a little tricky at first, but don't worry, we'll use our knowledge of arithmetic sequences and that handy formula we just learned to crack this. The key here is to figure out the first term ('a') and the common difference ('d'). Once we have those, finding the 10th term will be a piece of cake.
Step 1: Setting up the Equations
Let's use the formula for the nth term (an = a + (n - 1)d) to set up some equations based on the information we're given.
We know that the 5th term (a5) is 8. So, plugging in n = 5, we get:
a5 = a + (5 - 1)d = 8
Simplifying this, we have:
a + 4d = 8 ...(Equation 1)
Similarly, we know that the 9th term (a9) is 20. Plugging in n = 9, we get:
a9 = a + (9 - 1)d = 20
Simplifying this, we have:
a + 8d = 20 ...(Equation 2)
Now we have two equations with two unknowns (a and d). This is a classic system of equations that we can solve using a couple of different methods. Let's use the elimination method, which is often the easiest way to solve these types of problems.
Step 2: Solving for the Common Difference (d)
To use the elimination method, we want to subtract one equation from the other in a way that eliminates one of the variables. In this case, we can subtract Equation 1 from Equation 2 because the 'a' terms will cancel out.
Subtracting Equation 1 (a + 4d = 8) from Equation 2 (a + 8d = 20), we get:
(a + 8d) - (a + 4d) = 20 - 8
Simplifying this, we have:
4d = 12
Now, we can solve for 'd' by dividing both sides by 4:
d = 12 / 4
d = 3
Great! We've found the common difference, which is 3. This means that each term in the sequence is 3 more than the previous term. Now we just need to find the first term ('a').
Step 3: Solving for the First Term (a)
Now that we know the common difference (d = 3), we can plug it back into either Equation 1 or Equation 2 to solve for the first term ('a'). Let's use Equation 1 (a + 4d = 8) because it looks a little simpler.
Substituting d = 3 into Equation 1, we get:
a + 4(3) = 8
Simplifying this, we have:
a + 12 = 8
Now, we can solve for 'a' by subtracting 12 from both sides:
a = 8 - 12
a = -4
Awesome! We've found the first term, which is -4. So now we know that our arithmetic sequence starts with -4 and has a common difference of 3.
Step 4: Finding the 10th Term
We're almost there! Now that we know the first term (a = -4) and the common difference (d = 3), we can use the formula for the nth term to find the 10th term (a10).
Using the formula an = a + (n - 1)d, and plugging in n = 10, a = -4, and d = 3, we get:
a10 = -4 + (10 - 1)3
Simplifying this, we have:
a10 = -4 + (9)3
a10 = -4 + 27
a10 = 23
Solution: The 10th Term is 23
Therefore, the 10th term of the arithmetic sequence is 23!
We did it! We successfully found the 10th term of the arithmetic sequence by using the formula for the nth term, setting up a system of equations, and solving for the first term and common difference. This is a great example of how we can use mathematical tools to solve real problems.
Key Takeaways
Let's recap the key steps we took to solve this problem:
- Understand the Definition: We started by making sure we understood what an arithmetic sequence is and the meaning of common difference.
- Use the Formula: We used the formula for the nth term of an arithmetic sequence (an = a + (n - 1)d) to set up equations.
- Set up Equations: We created two equations based on the given information (the 5th term and the 9th term).
- Solve for 'd': We used the elimination method to solve for the common difference ('d').
- Solve for 'a': We plugged the value of 'd' back into one of the equations to solve for the first term ('a').
- Find the 10th Term: Finally, we used the formula for the nth term again, plugging in the values of 'a', 'd', and n = 10 to find the 10th term.
Practice Makes Perfect
The best way to get comfortable with arithmetic sequences is to practice solving problems. Try changing the given information in this problem (e.g., use different terms or different values) and see if you can still find the 10th term. You can also find plenty of practice problems online or in textbooks.
Remember, math is like learning a new language. The more you practice, the more fluent you'll become. Keep exploring, keep questioning, and keep learning! You got this!
Further Exploration
If you're interested in learning more about sequences and series, you can explore these topics:
- Geometric Sequences: Sequences where each term is multiplied by a constant value (instead of adding a constant value like in arithmetic sequences).
- Series: The sum of the terms in a sequence.
- Summation Notation: A shorthand way to represent the sum of a series.
These topics build upon the concepts we've learned today and can open up a whole new world of mathematical exploration. Keep the curiosity burning!