Arithmetic Sequence Formula Explained

An arithmetic sequence is a series of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference, often denoted by the letter 'd'. Understanding how to find the explicit formula for an arithmetic sequence is crucial for predicting any term in the sequence without having to calculate all the preceding terms. The explicit formula allows us to directly compute the value of the n-th term ($a_n$) using its position (n) in the sequence. This makes it a powerful tool in mathematics, especially in areas like algebra, calculus, and even in financial mathematics for calculating growth or decay over time. Let's dive into how we can derive and use this formula with a common example.

Understanding the Explicit Formula

The explicit formula for an arithmetic sequence is a mathematical expression that defines the n-th term ($a_n$) of the sequence directly in terms of 'n' and the first term ($a_1$) and the common difference (d). The standard form of this formula is $a_n = a_1 + (n-1)d$. Here's a breakdown of what each part represents:

• $a_n$: This is the term you want to find. It represents the value of the n-th term in the sequence.
• $a_1$: This is the first term of the sequence. It's the starting point of our series.
• n: This represents the position of the term in the sequence. For example, if you want to find the 5th term, then n = 5.
• d: This is the common difference, the constant value added to get from one term to the next.

To find the explicit formula for a given arithmetic sequence, you first need to identify the first term ($a_1$) and the common difference (d). Once you have these two values, you can substitute them into the general formula to create the specific explicit formula for that sequence. This formula is incredibly useful because it bypasses the need for step-by-step calculations, offering a direct route to any term, no matter how far down the sequence it is. It's like having a shortcut that saves you time and effort, making complex problems much more manageable.

Deriving the Formula for a Specific Sequence

Let's take the example sequence: **$8, 17, 26, 35, oldsymbol{ ext{ extellipsis}} $**. Our goal is to find the explicit formula that describes this sequence. First, we need to identify the key components: the first term ($a_1$) and the common difference (d).

• Identifying the First Term ($a_1$): The very first number in the sequence is 8. So, $a_1 = 8$.

• Identifying the Common Difference (d): To find the common difference, we subtract any term from its succeeding term. Let's check a few pairs:

  • $17 - 8 = 9$
  • $26 - 17 = 9$
  • $35 - 26 = 9$

The difference is consistently 9. Therefore, the common difference, $d = 9$.

Now that we have $a_1 = 8$ and $d = 9$, we can substitute these values into the general explicit formula: $a_n = a_1 + (n-1)d$.

Substituting our values, we get:

$a_n = 8 + (n-1)9$

This formula can be further simplified by distributing the 9:

$a_n = 8 + 9n - 9$

$a_n = 9n - 1$

However, the question asks for the formula in a specific format, matching one of the given options. Let's re-examine the options provided and see which one matches our derived formula or our intermediate step before simplification.

We found our explicit formula to be $a_n = 8 + (n-1)9$. Let's compare this to the given choices:

A. $a_n=9+8(n-1)$ - This has the first term and common difference swapped. B. $a_n=9+(n-1)$ - This has the wrong first term and common difference. C. $a_n=35+9(n-1)$ - This uses the fourth term (35) as the 'first term' and is structured incorrectly for the general formula. D. $a_n=8+9(n-1)$ - This directly matches our derived formula $a_n = 8 + (n-1)9$.

Therefore, the correct explicit formula for the arithmetic sequence $8, 17, 26, 35, oldsymbol ext{ extellipsis}} $ is $a_n = 8 + 9(n-1)$. This formula allows us to find any term in the sequence. For instance, to find the 100th term, we would simply plug in $n=100$ $a_{100 = 8 + 9(100-1) = 8 + 9(99) = 8 + 891 = 899$. This demonstrates the power and efficiency of the explicit formula.

Why Other Options Are Incorrect

It's important to understand why the other options provided are not the correct explicit formula for the sequence $8, 17, 26, 35, oldsymbol{ ext{ extellipsis}} $. Each incorrect option usually stems from a misunderstanding of which part of the formula represents the first term ($a_1$) and which represents the common difference (d), or how 'n' and '(n-1)' are used.

• Option A: $a_n = 9 + 8(n-1)$ This formula implies that the first term ($a_1$) is 9 and the common difference (d) is 8. Let's test this. If $a_1 = 9$ and $d = 8$, the sequence would start with 9, then $9+8=17$, then $17+8=25$, and so on. The sequence would be $9, 17, 25, 33, oldsymbol{ ext{ extellipsis}} $. This does not match our given sequence, which starts with 8 and has a common difference of 9. The values for $a_1$ and $d$ are incorrectly assigned here.

• Option B: $a_n = 9 + (n-1)$ This formula implies that the first term ($a_1$) is 9 and the common difference (d) is 1 (since $(n-1)$ is the same as $1 imes (n-1)$). If $a_1 = 9$ and $d = 1$, the sequence would be $9, 10, 11, 12, oldsymbol{ ext{ extellipsis}} $. This is clearly not our target sequence. Both the first term and the common difference are incorrect.

• Option C: $a_n = 35 + 9(n-1)$ This option uses the correct common difference (d=9), which is a good sign. However, it uses 35 as the 'first term'. In the formula $a_n = a_1 + (n-1)d$, $a_1$ must be the actual first term of the sequence. Here, 35 is the fourth term ($a_4$) of our sequence. If we were to use this formula, let's see what the first term ($n=1$) would be: $a_1 = 35 + 9(1-1) = 35 + 9(0) = 35$. This confirms that this formula generates a sequence starting with 35. While it might generate the same common difference, it generates a completely different sequence starting from a different point. The structure requires $a_1$ to be the initial value.

• Option D: $a_n = 8 + 9(n-1)$ This formula uses $a_1 = 8$ and $d = 9$. Let's check the first few terms:

  • For $n=1$: $a_1 = 8 + 9(1-1) = 8 + 9(0) = 8$. (Correct)
  • For $n=2$: $a_2 = 8 + 9(2-1) = 8 + 9(1) = 8 + 9 = 17$. (Correct)
  • For $n=3$: $a_3 = 8 + 9(3-1) = 8 + 9(2) = 8 + 18 = 26$. (Correct)
  • For $n=4$: $a_4 = 8 + 9(4-1) = 8 + 9(3) = 8 + 27 = 35$. (Correct)

This option perfectly matches the given sequence, confirming it as the correct explicit formula.

Conclusion: Mastering Arithmetic Sequences

Understanding the explicit formula for an arithmetic sequence is a fundamental skill in mathematics. It provides a direct and efficient way to calculate any term in a sequence, saving time and preventing errors that can arise from sequential calculations. By identifying the first term ($a_1$) and the common difference (d), you can construct the formula $a_n = a_1 + (n-1)d$ tailored to your specific sequence. As we saw with the sequence $8, 17, 26, 35, oldsymbol{ ext{ extellipsis}} $, the correct explicit formula is $a_n = 8 + 9(n-1)$. This formula accurately predicts every term in the sequence, highlighting the power of algebraic representation in mathematics.

Practice is key to mastering these concepts. Work through various examples, identify the patterns, and apply the formula. Don't be afraid to experiment with different sequences and test your understanding. If you're looking to deepen your knowledge of sequences and series, resources like Khan Academy offer excellent explanations, practice problems, and video tutorials that can further solidify your understanding of arithmetic sequences and related mathematical concepts.