Finding Arithmetic Sequence Expression An Easy To Understand Guide

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Hey guys! Ever stumbled upon a sequence of numbers and felt like you're trying to crack a secret code? Well, arithmetic sequences might seem daunting at first, but trust me, they're super fun to figure out once you get the hang of it. In this article, we're going to break down how to find the expression that represents an arithmetic sequence, using a real example to make it crystal clear. So, buckle up, and let's dive into the world of numbers!

Understanding Arithmetic Sequences

Before we jump into solving the problem, let's quickly recap what arithmetic sequences are all about. An arithmetic sequence is simply a list of numbers where the difference between any two consecutive terms is constant. This constant difference is what we call the common difference. For example, in the sequence 2, 4, 6, 8, ..., the common difference is 2 because each term is obtained by adding 2 to the previous term.

The general form of an arithmetic sequence can be represented as:

a,a+d,a+2d,a+3d,…a, a + d, a + 2d, a + 3d, \ldots

Where:

  • a is the first term of the sequence.
  • d is the common difference.

To find the expression that represents an arithmetic sequence, we need to determine the first term (a) and the common difference (d). Once we have these two values, we can plug them into the general formula for the nth term of an arithmetic sequence, which is:

an=a+(n−1)da_n = a + (n - 1)d

Where:

  • a_n is the nth term of the sequence.
  • n is the position of the term in the sequence.

Identifying the First Term and Common Difference

Let's start by pinpointing the first term. The first term is simply the first number in the sequence. Easy peasy, right? Next up, the common difference is the amount we add (or subtract) to get from one term to the next. You can find it by subtracting any term from the term that follows it. It's like figuring out the constant step size in a numerical dance.

The General Formula for the nth Term

Now, let's talk about the general formula. This is our golden ticket to finding any term in the sequence without having to list them all out. It's like having a secret map that leads directly to the treasure. The formula looks like this:

an=a+(n−1)da_n = a + (n - 1)d

Where:

  • a_n is the nth term we're trying to find.
  • a is the first term.
  • n is the term number (like 1st, 2nd, 3rd, etc.).
  • d is the common difference.

This formula is super handy because it connects the position of a term in the sequence (n) with its value (a_n). So, if you want to find the 100th term, you just plug in n = 100 and do the math!

Why This Formula Works

Ever wonder why this formula works? It's actually pretty intuitive. Think of it this way: You start with the first term (a). To get to the second term, you add the common difference (d) once. To get to the third term, you add the common difference twice (2d), and so on. So, to get to the nth term, you add the common difference (n - 1) times. That's exactly what the formula says!

Problem: Finding the Expression for the Sequence 15, 9, 3, -3, -9, ...

Now, let's tackle the problem at hand. We're given the sequence 15, 9, 3, -3, -9, ... and our mission is to find the expression that represents this arithmetic sequence. This means we need to find a formula that will give us any term in the sequence if we plug in its position (n). Let's break it down step by step.

Step 1 Identify the First Term (a)

The first term, denoted as a, is simply the first number in the sequence. In this case, the first term is 15. So, we have:

a=15a = 15

This is our starting point, the initial value from which the sequence begins its numerical journey. Think of it as the foundation upon which the rest of the sequence is built.

Step 2: Determine the Common Difference (d)

The common difference, denoted as d, is the constant value that is added (or subtracted) to each term to get the next term. To find the common difference, we can subtract any term from the term that follows it. Let's subtract the second term (9) from the first term (15):

d=9−15=−6d = 9 - 15 = -6

We can verify this by checking other pairs of consecutive terms:

  • 3 - 9 = -6
  • -3 - 3 = -6
  • -9 - (-3) = -6

The common difference is indeed -6. This means that each term in the sequence is 6 less than the previous term. The negative sign indicates that the sequence is decreasing.

Step 3 Apply the Formula for the nth Term

Now that we have the first term (a = 15) and the common difference (d = -6), we can plug these values into the formula for the nth term of an arithmetic sequence:

an=a+(n−1)da_n = a + (n - 1)d

Substituting the values, we get:

an=15+(n−1)(−6)a_n = 15 + (n - 1)(-6)

Step 4 Simplify the Expression

To find the expression, we need to simplify the equation. Let's distribute the -6:

an=15−6(n−1)a_n = 15 - 6(n - 1)

Now, let's distribute the -6 across the parentheses:

an=15−6n+6a_n = 15 - 6n + 6

Combine the constant terms:

an=21−6na_n = 21 - 6n

But wait! This simplified form isn't one of the options given. Let’s backtrack and see if we can match one of the provided choices. Going back to our equation before simplification:

an=15+(n−1)(−6)a_n = 15 + (n - 1)(-6)

Which can also be written as:

an=15−6(n−1)a_n = 15 - 6(n - 1)

Aha! This matches option D.

The Answer

Therefore, the expression that represents the arithmetic sequence 15, 9, 3, -3, -9, ... is:

D. $15-6(n-1)$

Breaking Down the Options

Let's take a quick look at why the other options are incorrect. This can help you understand the process even better and avoid common mistakes.

  • A. $15 + 6n$
    • This option has a positive 6n, which means the sequence would be increasing. Our sequence is decreasing, so this is incorrect.
  • B. $15 - 6n$
    • This option looks close, but if we plug in n = 1, we get 15 - 6(1) = 9, which is not the first term. So, this is also incorrect.
  • C. $15 + 6(n - 1)$
    • Similar to option A, this has a positive 6(n - 1), indicating an increasing sequence. This doesn't match our decreasing sequence.

By understanding why these options are wrong, you reinforce your understanding of how to find the correct expression.

Real-World Applications of Arithmetic Sequences

So, you might be wondering,