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Learn arithmetic sequences with worked examples, formulas, and real-world applications. Learn to find general terms and solve sequence problems.
---
## 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 $$b$$.
Visible text: 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 .
## Examples of Arithmetic Sequences
Consider the following number sequence:
$$4, 6, 8, 10, ...$$
In this sequence, we can see that:
- The difference between the second and first term: $$6 - 4 = 2$$
- The difference between the third and second term: $$8 - 6 = 2$$
- The difference between the fourth and third term: $$10 - 8 = 2$$
Visible text: - The difference between the second and first term:
- The difference between the third and second term:
- The difference between the fourth and third term:
Since the difference between consecutive terms is always $$2$$, this sequence is an arithmetic sequence with common difference $$b = 2$$.
Visible text: Since the difference between consecutive terms is always , this sequence is an arithmetic sequence with common difference .
## Common Difference in Arithmetic Sequences
The common difference ($$b$$) in an arithmetic sequence can be calculated by subtracting consecutive terms:
Visible text: The common difference () in an arithmetic sequence can be calculated by subtracting consecutive terms:
```math
b = U_2 - U_1 = U_3 - U_2 = U_4 - U_3 = ... = U_n - U_{n-1}
```
Where:
- $$U_n$$ represents the $$n$$
th term
- $$U_{n-1}$$ represents the ($$n-1$$
)th term
Visible text: - represents the
th term
- represents the (
)th term
## Formula for the nth Term
### General Formula
To determine the $$n$$th term of an arithmetic sequence, we can use the formula:
Visible text: To determine the th term of an arithmetic sequence, we can use the formula:
```math
U_n = a + (n-1)b
```
Where:
- $$U_n$$ = the $$n$$
th term
- $$a$$ = first term
- $$n$$ = term number
- $$b$$ = common difference
Visible text: - = the
th term
- = first term
- = term number
- = common difference
### Deriving the Formula
If we write the first few terms of an arithmetic sequence:
- $$1$$st term: $$U_1 = a$$
- $$2$$nd term: $$U_2 = a + b$$
- $$3$$rd term: $$U_3 = a + 2b$$
- $$4$$th term: $$U_4 = a + 3b$$
- $$5$$th term: $$U_5 = a + 4b$$
Visible text: - st term:
- nd term:
- rd term:
- th term:
- th term:
From this pattern, we can see that the $$n$$th term is:
Visible text: From this pattern, we can see that the th term is:
```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 $$1$$ has $$20$$ seats.
- Row $$2$$ has $$24$$ seats.
- Row $$3$$ has $$28$$ seats.
- Row $$4$$ has $$32$$ seats.
- Row $$5$$ has $$36$$ seats.
Visible text: - Row has seats.
- Row has seats.
- Row has seats.
- Row has seats.
- Row has 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
Visible text: **Step** : Finding the common difference between rows
- $$\text{Row }2 - \text{row }1$$: $$24 - 20 = 4$$
- $$\text{Row }3 - \text{row }2$$: $$28 - 24 = 4$$
- $$\text{Row }4 - \text{row }3$$: $$32 - 28 = 4$$
- $$\text{Row }5 - \text{row }4$$: $$36 - 32 = 4$$
Visible text: - :
- :
- :
- :
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:
Visible text: We can see that the difference between the number of seats in consecutive rows is . This means the number of seats in this theater forms an arithmetic sequence with:
- First term $$a = 20$$
- Common difference $$b = 4$$
Visible text: - First term
- Common difference
**Step** $$2$$: Using the formula to find the number of seats in row $$15$$
Visible text: **Step** : Using the formula to find the number of seats in row
Component: MathContainer
Children:
```math
U_{15} = a + (n-1)b
```
```math
U_{15} = 20 + (15-1)4
```
```math
U_{15} = 20 + (14)4
```
```math
U_{15} = 20 + 56
```
```math
U_{15} = 76
```
Therefore, there are $$76$$ seats in row $$15$$.
Visible text: Therefore, there are seats in row .
## First Exercise
Given an arithmetic sequence with the $$3$$rd term equal to $$9$$ and the $$6$$th term equal to $$18$$. Find the formula for the $$n$$th term.
Visible text: Given an arithmetic sequence with the rd term equal to and the th term equal to . Find the formula for the th term.
### Solution to First Exercise
To determine the formula for the general term, we need to find the values of the first term $$(a)$$ and the common difference $$(b)$$.
Visible text: To determine the formula for the general term, we need to find the values of the first term and the common difference .
Component: MathContainer
Children:
```math
U_3 = a + 2b = 9
```
```math
U_6 = a + 5b = 18
```
We eliminate these equations to find the value of $$b$$:
Visible text: We eliminate these equations to find the value of :
Component: MathContainer
Children:
```math
U_6 - U_3 = (a + 5b) - (a + 2b)
```
```math
18 - 9 = 3b
```
```math
9 = 3b
```
```math
b = 3
```
After finding $$b$$, we substitute it into the first equation to find $$a$$:
Visible text: After finding , we substitute it into the first equation to find :
Component: MathContainer
Children:
```math
a + 2b = 9
```
```math
a + 2(3) = 9
```
```math
a + 6 = 9
```
```math
a = 9 - 6 = 3
```
With $$a = 3$$ and $$b = 3$$, we can formulate the $$n$$th term:
Visible text: With and , we can formulate the th term:
Component: MathContainer
Children:
```math
U_n = a + (n-1)b
```
```math
U_n = 3 + (n-1)3
```
```math
U_n = 3 + 3n - 3
```
```math
U_n = 3n
```
Therefore, the formula for the general term of this sequence is $$U_n = 3n$$
Visible text: Therefore, the formula for the general term of this sequence is
## Second Exercise
Rudi saves money in a bank with a constant monthly increase. If in the $$5$$th month, he saves $$\text{Rp}70{,}000.00$$ and in the $$9$$th month, Rudi saves $$\text{Rp}90{,}000.00$$.
Visible text: Rudi saves money in a bank with a constant monthly increase. If in the th month, he saves and in the th month, Rudi saves .
1. What is the monthly increase in savings amount?
2. How much money did Rudi save for the first time?
### Solution to Second Exercise
Rudi's savings form an arithmetic sequence because the monthly increase is constant.
1. **Finding the monthly increase in savings**
```math
U_5 = 70{,}000
```
```math
a + (5-1)b = 70{,}000
```
```math
a + 4b = 70{,}000 \text{ ... (equation 1)}
```
```math
U_9 = 90{,}000
```
```math
a + (9-1)b = 90{,}000
```
```math
a + 8b = 90{,}000 \text{ ... (equation 2)}
```
Eliminating equations $$1$$ and $$2$$:
```math
4b = 20{,}000
```
```math
b = 5{,}000
```
Therefore, the monthly increase in Rudi's savings is $$\text{Rp}5{,}000.00$$.
2. **Finding Rudi's initial savings**
We have found $$b = 5{,}000$$, now we substitute it into equation $$(1)$$ to find $$a$$:
```math
a + 4b = 70{,}000
```
```math
a + 4(5{,}000) = 70{,}000
```
```math
a + 20{,}000 = 70{,}000
```
```math
a = 70{,}000 - 20{,}000
```
```math
a = 50{,}000
```
Therefore, Rudi's initial savings was $$\text{Rp}50{,}000.00$$.
Visible text: 1. **Finding the monthly increase in savings**
Eliminating equations and :
Therefore, the monthly increase in Rudi's savings is .
2. **Finding Rudi's initial savings**
We have found , now we substitute it into equation to find :
Therefore, Rudi's initial savings was .