# Nakafa Framework: LLM

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

Output docs content for large language models.

---

export const metadata = {
  title: "Arithmetic Sequence",
  description: "Master arithmetic sequences with step-by-step examples, formulas, and real-world applications. Learn to find nth terms and solve sequence problems.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/05/2025",
  subject: "Sequence and Series",
};

## Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is always constant. This difference in an arithmetic sequence is denoted by <InlineMath math="b" />.

## Examples of Arithmetic Sequences

Consider the following number sequence:

<InlineMath math="4, 6, 8, 10, ..." />

In this sequence, we can see that:

- The difference between the second and first term: <InlineMath math="6 - 4 = 2" />
- The difference between the third and second term: <InlineMath math="8 - 6 = 2" />
- The difference between the fourth and third term: <InlineMath math="10 - 8 = 2" />

Since the difference between consecutive terms is always 2, this sequence is an arithmetic sequence with common difference <InlineMath math="b = 2" />.

## Common Difference in Arithmetic Sequences

The common difference (<InlineMath math="b" />) in an arithmetic sequence can be calculated by subtracting consecutive terms:

<BlockMath math="b = U_2 - U_1 = U_3 - U_2 = U_4 - U_3 = ... = U_n - U_{n-1}" />

Where:

- <InlineMath math="U_n" /> represents the <InlineMath math="n" />
  th term
- <InlineMath math="U_{n-1}" /> represents the (<InlineMath math="n-1" />
  )th term

## Formula for the nth Term

### General Formula

To determine the <InlineMath math="n" />th term of an arithmetic sequence, we can use the formula:

<BlockMath math="U_n = a + (n-1)b" />

Where:

- <InlineMath math="U_n" /> = the <InlineMath math="n" />
  th term
- <InlineMath math="a" /> = first term
- <InlineMath math="n" /> = term number
- <InlineMath math="b" /> = common difference

### Deriving the Formula

If we write the first few terms of an arithmetic sequence:

- 1st term: <InlineMath math="U_1 = a" />
- 2nd term: <InlineMath math="U_2 = a + b" />
- 3rd term: <InlineMath math="U_3 = a + 2b" />
- 4th term: <InlineMath math="U_4 = a + 3b" />
- 5th term: <InlineMath math="U_5 = a + 4b" />

From this pattern, we can see that the <InlineMath math="n" />th term is:

<BlockMath math="U_n = a + (n-1)b" />

## Applications of Arithmetic Sequences

### Performing Arts Theater

Consider the number of seats in a performing arts theater:

- Row 1 = 20 seats
- Row 2 = 24 seats
- Row 3 = 28 seats
- Row 4 = 32 seats
- Row 5 = 36 seats

To determine the number of seats in a specific row, we need to find the pattern in this data.

**Step 1: Finding the common difference between rows**

- Row 2 - Row 1: <InlineMath math="24 - 20 = 4" />
- Row 3 - Row 2: <InlineMath math="28 - 24 = 4" />
- Row 4 - Row 3: <InlineMath math="32 - 28 = 4" />
- Row 5 - Row 4: <InlineMath math="36 - 32 = 4" />

We can see that the difference between the number of seats in consecutive rows is 4. This means the number of seats in this theater forms an arithmetic sequence with:

- First term (<InlineMath math="a" />) = 20
- Common difference (<InlineMath math="b" />) = 4

**Step 2: Using the formula to find the number of seats in row 15**

<div className="flex flex-col gap-4">
  <BlockMath math="U_{15} = a + (n-1)b" />
  <BlockMath math="U_{15} = 20 + (15-1)4" />
  <BlockMath math="U_{15} = 20 + (14)4" />
  <BlockMath math="U_{15} = 20 + 56" />
  <BlockMath math="U_{15} = 76" />
</div>

Therefore, there are 76 seats in row 15.

## Exercise 1

Given an arithmetic sequence with the 3rd term = 9 and the 6th term = 18. Find the formula for the nth term.

### Solution to Exercise 1

To determine the formula for the nth term, we need to find the values of the first term <InlineMath math="(a)" /> and the common difference <InlineMath math="(b)" />.

<div className="flex flex-col gap-4">
  <BlockMath math="U_3 = a + 2b = 9" />
  <BlockMath math="U_6 = a + 5b = 18" />
</div>

We eliminate these equations to find the value of b:

<div className="flex flex-col gap-4">
  <BlockMath math="U_6 - U_3 = (a + 5b) - (a + 2b)" />
  <BlockMath math="18 - 9 = 3b" />
  <BlockMath math="9 = 3b" />
  <BlockMath math="b = 3" />
</div>

After finding b, we substitute it into the first equation to find a:

<div className="flex flex-col gap-4">
  <BlockMath math="a + 2b = 9" />
  <BlockMath math="a + 2(3) = 9" />
  <BlockMath math="a + 6 = 9" />
  <BlockMath math="a = 9 - 6 = 3" />
</div>

With a = 3 and b = 3, we can formulate the nth term:

<div className="flex flex-col gap-4">
  <BlockMath math="U_n = a + (n-1)b" />
  <BlockMath math="U_n = 3 + (n-1)3" />
  <BlockMath math="U_n = 3 + 3n - 3" />
  <BlockMath math="U_n = 3n" />
</div>

Therefore, the formula for the nth term of this sequence is <InlineMath math="U_n = 3n" />

## Exercise 2

Rudi saves money in a bank with a constant monthly increase. If in the 5th month, he saves Rp70,000.00 and in the 9th month, Rudi saves Rp90,000.00.

a. What is the monthly increase in savings amount?

b. How much money did Rudi save for the first time?

### Solution to Exercise 2

Rudi's savings form an arithmetic sequence because the monthly increase is constant.

**a. Finding the monthly increase in savings**

<div className="flex flex-col gap-4">
  <BlockMath math="U_5 = 70,000" />
  <BlockMath math="a + (5-1)b = 70,000" />
  <BlockMath math="a + 4b = 70,000 \text{ ... (equation 1)}" />

  <BlockMath math="U_9 = 90,000" />
  <BlockMath math="a + (9-1)b = 90,000" />
  <BlockMath math="a + 8b = 90,000 \text{ ... (equation 2)}" />
</div>

Eliminating equations 1 and 2:

<div className="flex flex-col gap-4">
  <BlockMath math="4b = 20,000" />
  <BlockMath math="b = 5,000" />
</div>

Therefore, the monthly increase in Rudi's savings is Rp5,000.00.

**b. Finding Rudi's initial savings**

We have found b = 5,000, now we substitute it into equation 1 to find a:

<div className="flex flex-col gap-4">
  <BlockMath math="a + 4b = 70,000" />
  <BlockMath math="a + 4(5,000) = 70,000" />
  <BlockMath math="a + 20,000 = 70,000" />
  <BlockMath math="a = 70,000 - 20,000" />
  <BlockMath math="a = 50,000" />
</div>

Therefore, Rudi's initial savings was Rp50,000.00.
