Arithmetic Sequence: Find The Nth Term Formula

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Hey guys! Today, we're diving into the fascinating world of arithmetic sequences. Specifically, we're going to figure out how to find the formula for the nth term of a sequence. Let's take the arithmetic sequence 6, 10, 14, 18 as an example. Buckle up, because we're about to make math super easy and fun!

Understanding Arithmetic Sequences

Before we jump into finding the formula, let's make sure we all know what an arithmetic sequence actually is. An arithmetic sequence is simply a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted as 'd'.

Take a look at our sequence: 6, 10, 14, 18...

Can you spot the common difference? It's the number we add to each term to get the next one. In this case, we add 4 to 6 to get 10, add 4 to 10 to get 14, and so on. So, our common difference, d, is 4. Got it? Great!

Why is this important? Because understanding the common difference is key to unlocking the formula for the nth term. It's the heartbeat of the sequence, the rhythm that keeps it going. Without identifying this common difference, we'd be lost in the mathematical wilderness. So, always start by finding 'd'.

Now, let's consider the first term. We usually call this 'a' or 'a1'. In our sequence, the first term is 6. This is our starting point, the foundation upon which the rest of the sequence is built. Keep this value in mind, as we'll be using it in our formula.

In summary:

  • An arithmetic sequence has a constant difference between terms.
  • The common difference is denoted as 'd'.
  • The first term is denoted as 'a' or 'a1'.

Identifying 'a' and 'd'

Alright, let's solidify our understanding by pinpointing 'a' and 'd' in our given sequence: 6, 10, 14, 18. As we mentioned earlier, 'a' is the first term, which is undeniably 6. So, a = 6. Easy peasy, right?

Now for the common difference, 'd'. To find 'd', we subtract any term from the term that follows it. For instance, we can subtract 6 from 10, 10 from 14, or 14 from 18. Each of these subtractions will give us the same result: 4. Therefore, d = 4.

Let's double-check:

  • 10 - 6 = 4
  • 14 - 10 = 4
  • 18 - 14 = 4

Yep, it checks out! The common difference is indeed 4. Identifying 'a' and 'd' correctly is crucial because these values will be plugged into our formula to find the nth term. A mistake here would throw off our entire calculation, so accuracy is key.

Think of 'a' and 'd' as the coordinates to a hidden treasure. 'a' gives you the starting location, and 'd' provides the direction to follow. With both coordinates in hand, you're guaranteed to find the treasure, which in our case is the nth term formula.

The Nth Term Formula

Okay, now for the grand reveal: the formula for finding the nth term (often written as an) of an arithmetic sequence. Here it is:

an = a + (n - 1)d

Where:

  • an is the nth term we want to find.
  • a is the first term of the sequence.
  • n is the position of the term in the sequence (e.g., 1st, 2nd, 3rd, etc.).
  • d is the common difference.

This formula is your golden ticket to finding any term in the arithmetic sequence without having to list out all the terms before it. It's like having a mathematical GPS that can take you directly to the term you're looking for. Isn't that awesome?

Let's break it down a bit:

  • The (n - 1) part tells us how many times we need to add the common difference to the first term to reach the nth term. For example, to find the 5th term, we need to add the common difference 4 times to the first term.

  • Multiplying (n - 1) by d gives us the total amount we need to add to the first term to get to the nth term.

  • Adding this total to the first term, 'a', gives us the value of the nth term, an.

This formula is the cornerstone of working with arithmetic sequences. It's so important that you should memorize it, understand it, and learn to love it! With this formula in your toolkit, you'll be able to tackle any arithmetic sequence problem that comes your way.

Applying the Formula to Our Sequence

Alright, let's put this knowledge into action! We know that for our sequence (6, 10, 14, 18):

  • a = 6 (the first term)
  • d = 4 (the common difference)

Now, we plug these values into our formula:

an = a + (n - 1)d

an = 6 + (n - 1)4

That's it! We've successfully applied the formula. But let's simplify it further to make it look even cleaner. We distribute the 4:

an = 6 + 4n - 4

And then combine like terms:

an = 4n + 2

Boom! This is the formula for the nth term of our arithmetic sequence. This formula allows us to calculate any term in the sequence directly. For example, if we want to find the 10th term, we simply substitute n = 10:

a10 = 4(10) + 2 = 40 + 2 = 42

Therefore, the 10th term of the sequence is 42. Pretty neat, huh? No need to manually add 4 repeatedly until we get to the 10th term. Our formula does all the work for us!

To recap:

  1. We identified 'a' and 'd' from the sequence.
  2. We plugged these values into the nth term formula.
  3. We simplified the formula to get an = 4n + 2.
  4. We used the formula to find the 10th term of the sequence.

Verifying the Formula

To ensure our formula an = 4n + 2 is correct, let's test it with a few known terms from our sequence: 6, 10, 14, 18. We'll plug in n = 1, 2, 3, and 4 to see if we get the corresponding terms.

  • For n = 1 (the first term):

    • a1 = 4(1) + 2 = 4 + 2 = 6 (Correct!)

  • For n = 2 (the second term):

    • a2 = 4(2) + 2 = 8 + 2 = 10 (Correct!)

  • For n = 3 (the third term):

    • a3 = 4(3) + 2 = 12 + 2 = 14 (Correct!)

  • For n = 4 (the fourth term):

    • a4 = 4(4) + 2 = 16 + 2 = 18 (Correct!)

Our formula works perfectly for all the terms we tested! This gives us confidence that it's indeed the correct formula for the nth term of our arithmetic sequence. Verifying the formula is an important step to avoid any errors and ensure accuracy.

Think of it like baking a cake. You follow the recipe (the formula), but you also taste the batter along the way to make sure it's coming out right. Verifying the formula is like tasting the batter – it ensures that your final product (the nth term) is delicious!

In conclusion:

  • We tested our formula with known terms.
  • The formula correctly calculated all the terms.
  • We can confidently say that an = 4n + 2 is the correct formula.

Conclusion

And there you have it, folks! We've successfully determined the formula for the nth term of the arithmetic sequence 6, 10, 14, 18. By understanding the concept of arithmetic sequences, identifying the first term ('a') and the common difference ('d'), and applying the nth term formula, we were able to find the formula an = 4n + 2. Plus, we even verified our formula to make sure it was spot-on!

So next time you encounter an arithmetic sequence, don't sweat it. Just remember the steps we've covered today, and you'll be able to find the formula for the nth term in no time. Happy calculating!