Arithmetic Sequence

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 $b$.

Examples of Arithmetic Sequences

Consider the following number sequence:

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: $10 - 8 = 2$

Since the difference between consecutive terms is always $2$, this sequence is an arithmetic sequence with common difference $b = 2$.

Common Difference in Arithmetic Sequences

The common difference ($b$) in an arithmetic sequence can be calculated by subtracting consecutive terms:

Where:

• $U_n$ represents the $n$ th term
• $U_{n-1}$ represents the ($n-1$ )th term

Formula for the nth Term

General Formula

To determine the $n$th term of an arithmetic sequence, we can use the formula:

Where:

• $U_n$ = the $n$ th term
• $a$ = first term
• $n$ = term number
• $b$ = 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: $U_5 = a + 4b$

From this pattern, we can see that the $n$th term is:

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.

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

• $\text{Row }2 - \text{row }1$: $24 - 20 = 4$
• $\text{Row }3 - \text{row }2$: $28 - 24 = 4$
• $\text{Row }4 - \text{row }3$: $32 - 28 = 4$
• $\text{Row }5 - \text{row }4$: $36 - 32 = 4$

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 $a = 20$
• Common difference $b = 4$

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

Therefore, there are $76$ seats in row $15$.

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.

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 $(b)$.

We eliminate these equations to find the value of $b$:

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

With $a = 3$ and $b = 3$, we can formulate the $n$th term:

Therefore, the formula for the general term of this sequence is $U_n = 3n$

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 $\text{Rp}90{,}000.00$.

a. What is the monthly increase in savings amount?

b. 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.

a. Finding the monthly increase in savings

Eliminating equations $1$ and $2$:

Therefore, the monthly increase in Rudi's savings is $\text{Rp}5{,}000.00$.

b. Finding Rudi's initial savings

We have found $b = 5{,}000$, now we substitute it into equation $(1)$ to find $a$:

Therefore, Rudi's initial savings was $\text{Rp}50{,}000.00$.