Arithmetic Sequence

The Same Difference between Consecutive Terms

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is always constant. This constant is the common difference, written $b$. To test whether a list of numbers is arithmetic, subtract each term from the one after it and check that every subtraction returns the same number.

Examples of Arithmetic Sequences

Each term in this sequence increases by $2$:

In this sequence:

• 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 term number $n$
• $U_{n-1}$ represents term number $n-1$

Formula for the nth Term

Two forms describe the same arithmetic sequence. The general formula returns any term from its position, using the first term and the common difference. The recursive form builds each term from the term before it.

General Term of an Arithmetic Sequence

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

Where:

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

The pattern gives the $n$th term:

Recursive Form

The same arithmetic sequence can be defined from its first term and the rule that produces the next term:

The explicit formula gives any term directly, while the recursive form builds the sequence one term at a time.

Applications of Arithmetic Sequences

Each application below gives a starting value and a constant change per step. In the theater example, the seat count per row grows by four seats, and the general formula returns the seat count for any row number.

Performing Arts Theater

The number of seats in this performing arts theater grows with each row:

• 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

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

Consecutive rows differ by $4$ seats. Therefore, the seat counts form 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 $15$

There are $76$ seats in row $15$.

This result describes only row $15$. Finding the total number of seats in the first fifteen rows requires the corresponding arithmetic series.

Increasing Decreasing and Constant Sequences

An arithmetic sequence increases when $b>0$, decreases when $b<0$, and remains constant when $b=0$. For example:

The sign of the common difference therefore describes the direction in which the sequence changes.

First Exercise

This problem gives two terms of an arithmetic sequence at known positions and asks for the general formula. Each position produces one equation, and solving the two equations together gives the first term and the common difference.

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:

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

As a check, substituting $n=3$ and $n=6$ must reproduce the two given terms:

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

1. What is the monthly increase in savings amount?

2. How much did Rudi deposit in the first month?

Solution to Second Exercise

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

1. Finding the monthly increase in savings

   $$U_5 = 70{,}000$$

   $$a + (5-1)b = 70{,}000$$

   $$a + 4b = 70{,}000 \text{ ... (equation 1)}$$

   $$U_9 = 90{,}000$$

   $$a + (9-1)b = 90{,}000$$

   $$a + 8b = 90{,}000 \text{ ... (equation 2)}$$

   Eliminating equations 1 and 2:

   $$4b = 20{,}000$$

   $$b = 5{,}000$$

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

2. Finding Rudi's first-month deposit

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

   $$a + 4b = 70{,}000$$

   $$a + 4(5{,}000) = 70{,}000$$

   $$a + 20{,}000 = 70{,}000$$

   $$a = 70{,}000 - 20{,}000$$

   $$a = 50{,}000$$

   Therefore, Rudi deposited $\text{Rp}50{,}000.00$ in the first month.