Nth Term Equation: Finding A₂₄ In Sequences

Hey guys! Let's dive into the exciting world of sequences and figure out how to find the equation for the nth term, and then, just for kicks, we'll find the 24th term (a₂₄) for some given sequences. This might sound intimidating, but trust me, we'll break it down step-by-step so it’s super easy to follow. So, grab your pencils and let's get started!

Understanding Arithmetic Sequences

Before we jump into finding the equations, let's quickly recap what an arithmetic sequence is. An arithmetic sequence is essentially a list of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference. For instance, in the sequence 2, 4, 6, 8, ..., the common difference is 2 because you add 2 to each term to get the next one. Understanding this basic concept is crucial because the method we use to find the nth term equation relies on this consistent pattern.

To find the equation for the nth term, we'll use a handy formula that's specifically designed for arithmetic sequences. This formula allows us to calculate any term in the sequence without having to list out all the terms before it. Think of it like a shortcut that saves us a ton of time and effort. We'll go through examples shortly, but keep in mind that identifying the common difference is the first and most important step. This value will play a key role in our equation. Once we have the equation, finding a specific term, like a₂₄, becomes a piece of cake. We simply substitute 24 for 'n' in the equation, and voila, we have our answer!

So, remember, the key to tackling these problems is to first identify the common difference, then use the formula to build our equation, and finally, substitute the desired term number to find its value. With a little practice, you'll be finding nth terms like a pro in no time!

Sequence 1: 1, 3, 5, 7, ...

Okay, let’s kick things off with our first sequence: 1, 3, 5, 7, .... The first thing we need to do, as we discussed, is to identify the common difference. Look at the sequence. What number are we adding each time to get to the next term? If you said 2, you’re spot on! So, our common difference (let's call it 'd') is 2. Now that we have this crucial piece of information, we can start building our equation for the nth term.

The general formula for the nth term (aₙ) of an arithmetic sequence is: aₙ = a₁ + (n - 1)d, where a₁ is the first term of the sequence and 'd' is the common difference. In our sequence, the first term (a₁) is 1, and we’ve already determined that 'd' is 2. Let’s plug these values into our formula: aₙ = 1 + (n - 1)2. Now, let’s simplify this equation. Distribute the 2: aₙ = 1 + 2n - 2. Combine like terms: aₙ = 2n - 1. And there you have it! This is the equation that represents the nth term of the sequence 1, 3, 5, 7, ...

Now, for the fun part: finding a₂₄. This means we need to find the 24th term in the sequence. To do this, we simply substitute n = 24 into our equation: a₂₄ = 2(24) - 1. Let's solve it: a₂₄ = 48 - 1 = 47. So, the 24th term in the sequence 1, 3, 5, 7, ... is 47. See? It's not as scary as it looks! By following these steps – identifying the common difference, plugging values into the formula, and simplifying – we can confidently find the nth term and any specific term in the sequence.

Sequence 2: -1, -4, -7, -10, ...

Alright, let's move on to our next sequence: -1, -4, -7, -10, .... Remember our first step? That's right, we need to find the common difference. This time, the numbers are decreasing, so we know we're dealing with a negative common difference. To get from -1 to -4, we subtract 3. To get from -4 to -7, we also subtract 3. The same goes for -7 to -10. So, our common difference (d) is -3. Make sure to pay close attention to the signs – a negative common difference is just as important as a positive one!

Now that we have our common difference, we can use the same formula as before: aₙ = a₁ + (n - 1)d. In this sequence, the first term (a₁) is -1, and the common difference (d) is -3. Let's substitute these values into the formula: aₙ = -1 + (n - 1)(-3). Time to simplify! Distribute the -3: aₙ = -1 - 3n + 3. Combine like terms: aₙ = -3n + 2. Great! We've got our equation for the nth term of the sequence -1, -4, -7, -10, ...

Now, let's find a₂₄. Substitute n = 24 into our equation: a₂₄ = -3(24) + 2. Let's calculate: a₂₄ = -72 + 2 = -70. So, the 24th term in this sequence is -70. Notice how the negative common difference led to a negative value for the 24th term. This makes sense, as the sequence is constantly decreasing. The key takeaway here is to be meticulous with your calculations and always double-check your signs. You're doing awesome so far!

Sequence 3: -4, -9, -14, -19, ...

Let's keep the ball rolling with sequence number three: -4, -9, -14, -19, .... You know the drill by now, guys! Our first task is to identify the common difference. Looking at the sequence, we can see that the numbers are decreasing, indicating a negative common difference. To get from -4 to -9, we subtract 5. To go from -9 to -14, we also subtract 5, and so on. Therefore, our common difference (d) is -5.

Now that we've pinpointed the common difference, it's time to put our formula to work: aₙ = a₁ + (n - 1)d. In this sequence, the first term (a₁) is -4, and the common difference (d) is -5. Let's plug these values into our trusty formula: aₙ = -4 + (n - 1)(-5). Time for the simplification process! We distribute the -5: aₙ = -4 - 5n + 5. Then, we combine the like terms: aₙ = -5n + 1. Fantastic! We've successfully derived the equation for the nth term of the sequence -4, -9, -14, -19, ...

With the equation in hand, finding a₂₄ is a breeze. We substitute n = 24 into our equation: a₂₄ = -5(24) + 1. Let's crunch the numbers: a₂₄ = -120 + 1 = -119. So, the 24th term in this sequence is -119. Again, the negative common difference has led us to a negative value for the 24th term, which aligns with the decreasing nature of the sequence. Remember, consistency is key – keep applying the same steps, and you'll conquer any sequence that comes your way!

Sequence 4: 7, 13, 19, 25, ...

Last but not least, let's tackle our final sequence: 7, 13, 19, 25, .... By now, you should be feeling like pros at this! Let’s start with our usual first step: finding the common difference. This time, the numbers are increasing, so we're expecting a positive common difference. To get from 7 to 13, we add 6. To get from 13 to 19, we also add 6, and so on. So, our common difference (d) is 6.

Now, let's bring out our trusty formula: aₙ = a₁ + (n - 1)d. In this sequence, the first term (a₁) is 7, and the common difference (d) is 6. Let’s substitute these values into the formula: aₙ = 7 + (n - 1)6. Time for the simplification dance! Distribute the 6: aₙ = 7 + 6n - 6. Combine those like terms: aₙ = 6n + 1. Excellent! We've determined the equation for the nth term of the sequence 7, 13, 19, 25, ...

Time to find a₂₄! We substitute n = 24 into our equation: a₂₄ = 6(24) + 1. Let’s do the math: a₂₄ = 144 + 1 = 145. Therefore, the 24th term in this sequence is 145. This time, we have a positive value, which makes sense given the increasing nature of the sequence and the positive common difference. You've successfully navigated all four sequences! Give yourselves a pat on the back!

Conclusion

So, there you have it! We've walked through how to write an equation for the nth term of an arithmetic sequence and how to use that equation to find a specific term, like a₂₄. Remember, the key steps are identifying the common difference, plugging the first term and common difference into the formula aₙ = a₁ + (n - 1)d, simplifying the equation, and then substituting the value of 'n' to find the desired term.

With a little practice, you’ll be able to tackle any arithmetic sequence that comes your way. Keep practicing, and you'll become a sequence-solving superstar! You've got this! Remember, math is just like any other skill – the more you practice, the better you get. So, keep those pencils moving and keep exploring the amazing world of sequences and patterns. You're doing great, guys! Keep up the awesome work!