To find the sum of the first ( n ) terms in an arithmetic sequence, we first need to know what an arithmetic sequence is.

An arithmetic sequence is a list of numbers where the difference between each number and the next one is the same. This difference is called the common difference, and we use the letter ( d ) to represent it.

Important Parts of an Arithmetic Sequence

1. First Term: This is the very first number in the sequence, written as ( a_1 ).

2. Common Difference: This is the amount added to each term to get the next term. We write it as ( d = a_{n+1} - a_n ).

3. n-th Term: You can find any term in the sequence with this formula:

   $$a_n = a_1 + (n-1)d$$

How to Find the Sum of the First n Terms

If you want to find the sum of the first ( n ) terms, you can use this formula:

$$S_n = \frac{n}{2} (a_1 + a_n)$$

Here’s what the letters mean:

• ( S_n ) is the total of the first ( n ) terms,
• ( a_1 ) is the first term,
• ( a_n ) is the n-th term.

Understanding the Sum Formula

To get this formula, we look at how the terms relate to each other. For example, if we write out the first ( n ) terms:

$$S_n = a_1 + a_2 + a_3 + ... + a_n$$

Now, if we write this sum backward, we get:

$$S_n = a_n + a_{n-1} + a_{n-2} + ... + a_1$$

When we add these two versions of ( S_n ) together, we have:

$$2S_n = (a_1 + a_n) + (a_2 + a_{n-1}) + (a_3 + a_{n-2}) + ... + (a_n + a_1)$$

Each pair adds up to the same number ( (a_1 + a_n) ). There will be ( n ) pairs when ( n ) is even. If ( n ) is odd, there will be one middle term left over.

So, we get:

$$2S_n = n(a_1 + a_n)$$

Now, if we divide both sides by 2, we find:

$$S_n = \frac{n}{2} (a_1 + a_n)$$

Another Way to Find the Sum Using the Common Difference

You can also find the sum using the common difference. If we remember that ( a_n = a_1 + (n-1)d ), we can write the sum like this:

$$S_n = \frac{n}{2} (2a_1 + (n-1)d)$$

This is helpful if you already know the first term and the common difference, so you don’t have to find the n-th term first.

Example Calculation

Let’s look at an example where:

• The first term ( a_1 = 2 ),
• The common difference ( d = 3 ),
• And we want to find the sum of the first ( n = 5 ) terms.

1. First, calculate ( a_n ):

   $$a_5 = 2 + (5-1) \cdot 3 = 2 + 12 = 14$$

2. Now, use the sum formula:

   $$S_5 = \frac{5}{2} (2 + 14) = \frac{5}{2} \cdot 16 = 5 \cdot 8 = 40$$

Conclusion

Knowing how to calculate the sum of the first ( n ) terms in an arithmetic sequence is important in math. It helps you understand different series and improves your math skills. The formulas we talked about make it easier to find the sums clearly and correctly.