Finding The 18th Term Of The Sequence -6, -2, 2, 6
Hey guys! Ever stumbled upon a sequence of numbers that just seems to dance to its own rhythm? Well, let's dive into one such intriguing sequence: -6, -2, 2, 6, ... Our mission? To decipher the hidden rule governing this sequence and, more excitingly, to pinpoint the 18th term. Buckle up, because we're about to embark on a mathematical adventure!
Delving into the Number Sequence: -6, -2, 2, 6, ...
At first glance, this sequence might seem like a random jumble of numbers. But, as with any mathematical puzzle, there's a pattern lurking beneath the surface. Our initial task is to identify this pattern, the secret sauce that dictates how the sequence unfolds. Let's carefully examine the differences between consecutive terms:
- From -6 to -2: The difference is 4
- From -2 to 2: The difference is also 4
- From 2 to 6: And again, the difference is 4
Aha! A constant difference emerges. This consistent difference of 4 between successive terms is a telltale sign of an arithmetic sequence. An arithmetic sequence, for those unfamiliar, is a sequence where the difference between any two consecutive terms remains constant. This constant difference is affectionately known as the common difference.
Now that we've unmasked the sequence as an arithmetic one, we can leverage the power of arithmetic sequence formulas to our advantage. These formulas are the magic keys that will unlock the mystery of the 18th term. The most fundamental formula for an arithmetic sequence is the one that defines the nth term (often denoted as a_n):
a_n = a_1 + (n - 1) * d
Where:
- a_n is the nth term we're trying to find
- a_1 is the first term of the sequence
- n is the term number we're interested in (in our case, 18)
- d is the common difference
With this formula in our arsenal, we're well-equipped to tackle the challenge ahead. Let's move on to the next step: plugging in the values and calculating the elusive 18th term.
Unveiling the Rule: The Arithmetic Sequence Formula
Having established that our sequence is arithmetic, we now have the golden key to unlock its secrets: the arithmetic sequence formula. This formula, as we discussed earlier, is our guiding light in determining any term in the sequence. Let's recap the formula:
a_n = a_1 + (n - 1) * d
Where:
- a_n is the nth term we're seeking
- a_1 is the first term, which in our sequence is -6
- n is the term number, and we're hunting for the 18th term, so n = 18
- d is the common difference, which we've diligently identified as 4
Now, let's put these pieces together like a mathematical jigsaw puzzle. We substitute the known values into the formula:
a_18 = -6 + (18 - 1) * 4
Time for some arithmetic acrobatics! First, we simplify the expression inside the parentheses:
a_18 = -6 + (17) * 4
Next, we perform the multiplication:
a_18 = -6 + 68
And finally, the grand reveal! We add the numbers:
a_18 = 62
Eureka! We've successfully deciphered the 18th term of the sequence. It's a resounding 62! This demonstrates the power of understanding the underlying rule of a sequence and applying the appropriate formula. Let's take a moment to appreciate the elegance of mathematics, where patterns and formulas intertwine to reveal hidden truths.
Finding the 18th Term: A Step-by-Step Calculation
Alright, guys, let's get down to the nitty-gritty and calculate the 18th term. We've already laid the groundwork by identifying the sequence as arithmetic and unveiling the magic formula. Now, it's time to put that formula to work. As a quick refresher, the formula is:
a_n = a_1 + (n - 1) * d
We've also established that:
- a_1 (the first term) = -6
- n (the term number we want) = 18
- d (the common difference) = 4
Now, let's plug these values into the formula:
a_18 = -6 + (18 - 1) * 4
Let's break this down step by step to ensure we don't miss anything. First up, the parentheses: (18 - 1) = 17. So, our equation now looks like this:
a_18 = -6 + 17 * 4
Next, we tackle the multiplication: 17 * 4 = 68. Our equation is further simplified:
a_18 = -6 + 68
Finally, the grand finale! We perform the addition: -6 + 68 = 62. And there we have it!
a_18 = 62
So, the 18th term of the sequence -6, -2, 2, 6, ... is 62. We've successfully navigated the mathematical terrain and emerged victorious! It's always satisfying to see how a formula can be used to solve a problem, isn't it? We've not only found the answer but also reinforced our understanding of arithmetic sequences. Now, let's recap our journey and solidify our understanding.
Conclusion: The Beauty of Arithmetic Sequences
Guys, we've journeyed through the fascinating world of number sequences, specifically focusing on an arithmetic sequence. We started with a seemingly simple sequence: -6, -2, 2, 6, ... and our quest was to uncover its hidden rule and determine the 18th term. Through careful observation, we identified the constant difference of 4 between consecutive terms, which led us to the realization that this was indeed an arithmetic sequence.
We then introduced the powerhouse of arithmetic sequences: the formula a_n = a_1 + (n - 1) * d. This formula became our guiding star, allowing us to navigate the complexities of the sequence and pinpoint the 18th term. By plugging in the values we diligently extracted from the sequence – a_1 = -6, n = 18, and d = 4 – we embarked on a step-by-step calculation. We simplified, multiplied, and added our way to the grand finale: a_18 = 62.
Thus, we triumphantly concluded that the 18th term of the sequence is 62. This exercise highlights the elegance and power of mathematics. It showcases how a seemingly simple pattern, like an arithmetic sequence, can be described by a concise formula that allows us to predict any term in the sequence. More broadly, this exploration underscores the beauty of mathematical reasoning: the process of observing patterns, formulating rules, and applying those rules to solve problems.
So, the next time you encounter a number sequence, remember our adventure. Remember the thrill of uncovering the pattern, the satisfaction of applying the formula, and the joy of arriving at the solution. Keep exploring, keep questioning, and keep embracing the wonders of mathematics!