How to Find the nth Term of an Arithmetic Sequence Simply

Finding the nth term of an arithmetic sequence can seem tricky at first, but with a step-by-step approach, it becomes manageable. Let's break it down!

### Understanding Arithmetic Sequences
An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant, known as the common difference. For example, in the sequence -48, -96, -144, -192, we can see that each term decreases by 48.

### Step 1: Identify the First Term and Common Difference
In our example, the first term (a1) is -48, and the common difference (d) is -48 as well. This means:
- First term (a1) = -48
- Common difference (d) = -48

### Step 2: Use the Formula for the nth Term
The formula for finding the nth term (an) of an arithmetic sequence is:

\[ a_n = a_1 + (n - 1)d \]

Where:
- an = nth term
- a1 = first term
- d = common difference
- n = term number

### Step 3: Plug in Your Values
Now, let's plug in the values we identified:
- a1 = -48
- d = -48

So the formula becomes:
\[ a_n = -48 + (n - 1)(-48) \]

### Step 4: Simplify the Expression
Next, we simplify the expression:
1. Distributing the -48:
\[ a_n = -48 + (-48n + 48) \]
2. Combine like terms:
\[ a_n = -48n \]

### Step 5: Verify Your Formula
To ensure that our formula is correct, we can check some values:
- For n = 1:
\[ a_1 = -48(1) = -48 \] ✅
- For n = 2:
\[ a_2 = -48(2) = -96 \] ✅
- For n = 3:
\[ a_3 = -48(3) = -144 \] ✅

### Real-World Applications
Understanding arithmetic sequences is not only crucial for math classes but also has real-world applications. For instance, if you save a fixed amount of money each month, your total savings over time can be represented as an arithmetic sequence. Similarly, when calculating distances or steps taken, knowing how to find the nth term can help you track progress effectively.

### Conclusion
By identifying the first term and the common difference, you can easily find the nth term of any arithmetic sequence. This understanding will serve you well in various mathematical contexts and beyond!