# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/sequence-series/series-concept
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/sequence-series/series-concept/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Series Concept",
  description: "Understand series as sequential sums through handshake examples. Discover how adding sequence terms creates arithmetic and geometric series patterns.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/08/2025",
  subject: "Sequence and Series",
};

## Understanding Number Series

Have you ever thought about how to sum the terms of a number sequence? For example, adding 1 + 2 + 3 + ... and so on? Well, this is what's called a **number series**.

So, a **number series** is the result of sequentially adding the terms of a number sequence. Just like sequences, there are two main types of series: **arithmetic series** (the sum of terms in an arithmetic sequence) and **geometric series** (the sum of terms in a geometric sequence). But don't worry, we'll discuss the detailed formulas later.

Now, let's look at a real-world example of the series concept through a simple exploration.

## Number of Handshakes

Imagine there are several people in a group. If each person shakes hands exactly once with every other person in the group, how many total handshakes occur?

Let's try to count:

- **If there are 2 people:** There is only <InlineMath math="1" /> handshake.
- **If there are 3 people:** The first person shakes hands with 2 others. The second person has already shaken hands with the first, so they only need to shake hands with the third person. The total is <InlineMath math="2 + 1 = 3" /> handshakes.
- **If there are 4 people:** Following the same pattern, we get <InlineMath math="3 + 2 + 1 = 6" /> handshakes.
- **If there are 5 people:** The total is <InlineMath math="4 + 3 + 2 + 1 = 10" /> handshakes.

Notice the pattern! The number of handshakes forms a sequential sum of natural numbers.

| Number of people present | Number of handshakes | Breakdown of handshakes       |
| :----------------------- | :------------------- | :---------------------------- |
| Two people               | 1                    | 1                             |
| Three people             | 3                    | <InlineMath math="1+2" />     |
| Four people              | 6                    | <InlineMath math="1+2+3" />   |
| Five people              | 10                   | <InlineMath math="1+2+3+4" /> |

Summations like <InlineMath math="1+2+3+4" /> are examples of a **number series**. In this case, the series is formed from the sum of the terms of the sequence of natural numbers <InlineMath math="(1, 2, 3, 4, ...)" />.

### Does This Form a Sequence?

Interesting question: does the number of handshakes itself <InlineMath math="(1, 3, 6, 10, ...)" /> form an arithmetic or geometric sequence? The answer is no. This sequence does not have a constant difference or ratio between its terms (<InlineMath math="3-1=2" />, <InlineMath math="6-3=3" />, <InlineMath math="10-6=4" />).

However, the **breakdown** of the number of handshakes (<InlineMath math="1" />, <InlineMath math="1+2" />, <InlineMath math="1+2+3" />, <InlineMath math="1+2+3+4" />) clearly represents the sum of the terms of a sequence (specifically, the sequence of natural numbers). This is the core concept of a **series**.

So, this handshake exploration shows how the concept of a series arises from summing the terms of a pattern or number sequence in everyday situations.
