# Nakafa Framework: LLM

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

Output docs content for large language models.

---

export const metadata = {
  title: "Arithmetic Series",
  description: "Discover Gauss's brilliant method for calculating arithmetic series sums. Learn formulas, solve problems, and master sum calculations with examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/08/2025",
  subject: "Sequence and Series",
};

## Understanding Arithmetic Series

Ever heard the story about Carl Friedrich Gauss, the math genius? When he was in elementary school, his teacher assigned the task of summing all numbers from 1 to 100: <InlineMath math="1 + 2 + 3 + \dots + 98 + 99 + 100" />. The teacher hoped this would keep the students busy for a while.

But Gauss had a brilliant idea! He didn't sum them one by one. This sequential summation of terms from an _arithmetic sequence_ (a sequence with a constant difference between terms) is what we call an **Arithmetic Series**.

For example, <InlineMath math="1, 2, 3, \dots, 100" /> is an arithmetic sequence with the first term <InlineMath math="a=1" /> and a common difference <InlineMath math="b=1" />. The corresponding arithmetic series is <InlineMath math="1 + 2 + 3 + \dots + 100" />.

### How Did Gauss Calculate It?

Gauss noticed an interesting pattern:

- If the first term <InlineMath math="(1)" /> is added to the last term <InlineMath math="(100)" />, the result is <InlineMath math="101" />.
- If the second term <InlineMath math="(2)" /> is added to the second-to-last term <InlineMath math="(99)" />, the result is also <InlineMath math="101" />.
- If the third term <InlineMath math="(3)" /> is added to the third-to-last term <InlineMath math="(98)" />, the result is still <InlineMath math="101" />.
- This pattern continues!

<div className="flex flex-col mt-4">
  <BlockMath math="\underbrace{1+100}_{101}, \underbrace{2+99}_{101}, \underbrace{3+98}_{101}, \dots, \underbrace{50+51}_{101}" />
</div>

It turns out there are 50 pairs of numbers, each summing to 101. So, the total sum is <InlineMath math="50 \times 101 = 5050" />. Clever, right?

## Finding the General Formula

We can use Gauss's method to derive a general formula for the sum of the first <InlineMath math="n" /> terms of an arithmetic series, usually denoted by <InlineMath math="S_n" />.

Let's say we have an arithmetic series:

<BlockMath math="S_n = U_1 + U_2 + U_3 + \dots + U_{n-1} + U_n" />

If written out with the first term <InlineMath math="(a)" /> and the common difference <InlineMath math="(b)" />:

<BlockMath math="S_n = a + (a+b) + (a+2b) + \dots + (a+(n-2)b) + (a+(n-1)b)" />

Now, let's rewrite the series <InlineMath math="S_n" /> in reverse order, from the last term to the first:

<BlockMath math="S_n = U_n + U_{n-1} + \dots + U_2 + U_1" />
Or:
<BlockMath math="S_n = (a+(n-1)b) + (a+(n-2)b) + \dots + (a+b) + a" />

Next, let's add these two versions of <InlineMath math="S_n" /> together, term by term:

<div className="flex flex-col gap-4">
  <BlockMath math="S_n = a + (a+b) + \dots + (a+(n-2)b) + (a+(n-1)b)" />
  <BlockMath math="S_n = (a+(n-1)b) + (a+(n-2)b) + \dots + (a+b) + a" />
  <BlockMath math="2S_n = [2a+(n-1)b] + [2a+(n-1)b] + \dots + [2a+(n-1)b] + [2a+(n-1)b]" />
</div>

Notice! The sum of each pair of terms (top and bottom) is always the same, which is <InlineMath math="2a + (n-1)b" />. Since there are <InlineMath math="n" /> terms, there are <InlineMath math="n" /> such identical sums.

So, we get:

<BlockMath math="2S_n = n \times (2a + (n-1)b)" />

By dividing both sides by 2, we obtain the formula for the sum of the first <InlineMath math="n" /> terms of an arithmetic series:

<BlockMath math="S_n = \frac{n}{2} (2a + (n-1)b)" />

## Practical Formulas for Arithmetic Series

There are two main formulas commonly used to calculate <InlineMath math="S_n" />:

1.  If the **first term <InlineMath math="(a)" />** and the **common difference <InlineMath math="(b)" />** are known:

    <BlockMath math="S_n = \frac{n}{2}(2a + (n - 1)b)" />

2.  If the **first term <InlineMath math="(a)" />** and the **<InlineMath math="(n)" />-th term <InlineMath math="(U_n)" />** are known:
    Recall the formula for the <InlineMath math="n" />-th term is <InlineMath math="U_n = a + (n-1)b" />. Substituting this into the first formula:

    <div className="flex flex-col gap-4">
      <BlockMath math="S_n = \frac{n}{2}(a + a + (n - 1)b)" />
      <BlockMath math="S_n = \frac{n}{2}(a + U_n)" />
    </div>

    This second formula resembles Gauss's method: the sum of the first and last terms, multiplied by the number of pairs <InlineMath math="(\frac{n}{2})" />.

**Notation:**

- <InlineMath math="S_n" /> = Sum of the first <InlineMath math="n" /> terms
- <InlineMath math="n" /> = Number of terms
- <InlineMath math="a" /> = First term (<InlineMath math="U_1" />)
- <InlineMath math="b" /> = Common difference (difference between terms)
- <InlineMath math="U_n" /> = The <InlineMath math="n" />
  -th term

## Example Problems

### Problem 1

Recalculate the sum of the series <InlineMath math="1 + 2 + 3 + \dots + 98 + 99 + 100" />.

Given:

- First term <InlineMath math="a = 1" />
- Last term <InlineMath math="U_n = U_{100} = 100" />
- Number of terms <InlineMath math="n = 100" />

Since <InlineMath math="a" /> and <InlineMath math="U_n" /> are known, we use the second formula:

<div className="flex flex-col gap-4">
  <BlockMath math="S_n = \frac{n}{2}(a + U_n)" />
  <BlockMath math="S_{100} = \frac{100}{2}(1 + 100)" />
  <BlockMath math="S_{100} = 50(101)" />
  <BlockMath math="S_{100} = 5050" />
</div>

The result is exactly the same as Gauss's calculation!

### Problem 2

Given the arithmetic series: <InlineMath math="13 + 16 + 19 + 22 + \dots" />. Calculate the sum of the first 30 terms <InlineMath math="(S_{30})" />!

Given:

- First term <InlineMath math="a = 13" />
- Common difference <InlineMath math="b = 16 - 13 = 3" />
- Number of terms to sum <InlineMath math="n = 30" />

Since <InlineMath math="a" /> and <InlineMath math="b" /> are known, we use the first formula:

<div className="flex flex-col gap-4">
  <BlockMath math="S_n = \frac{n}{2}(2a + (n - 1)b)" />
  <BlockMath math="S_{30} = \frac{30}{2}[2(13) + (30 - 1)(3)]" />
  <BlockMath math="S_{30} = 15[26 + (29)(3)]" />
  <BlockMath math="S_{30} = 15(26 + 87)" />
  <BlockMath math="S_{30} = 15(113)" />
  <BlockMath math="S_{30} = 1695" />
</div>

So, the sum of the first 30 terms of this series is 1695.