Difference between Arithmetic and Geometric Sequences

Sequences

A number sequence is an arrangement of numbers that follows a specific pattern. Two common types are arithmetic sequences and geometric sequences. An arithmetic sequence adds a fixed value from one term to the next, while a geometric sequence multiplies by a fixed value, and that single difference decides which formulas apply.

Arithmetic Sequences

An arithmetic sequence moves from one term to the next by adding the same number every time. The definition below writes that rule, and the general-term formula returns any term from its position.

Definition of Arithmetic Sequences

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant.

If we have a sequence $a_1, a_2, a_3, ..., a_n$, then it is an arithmetic sequence if the difference between consecutive terms is always the same:

In this equation, $b$ is the constant difference between consecutive terms.

Formula for the General Term of an Arithmetic Sequence

For an arithmetic sequence with first term $a$ and common difference $b$, the formula for the general term is:

Geometric Sequences

A geometric sequence moves from one term to the next by multiplying by the same number every time. Ratios describe that rule, and the general-term formula again returns any term from its position.

Definition of Geometric Sequences

A geometric sequence is a sequence of numbers where the ratio between consecutive terms is constant.

If we have a sequence $a_1, a_2, a_3, ..., a_n$, then it is a geometric sequence if the ratio between consecutive terms is always the same:

In this equation, $r$ is the constant ratio. If a denominator is zero, the quotient form is unavailable, but the equivalent multiplication rule $a_{n+1}=ra_n$ remains meaningful.

Formula for the General Term of a Geometric Sequence

For a geometric sequence with first term $a$ and common ratio $r$, the formula for the general term is:

Comparing Arithmetic and Geometric Sequences

Both types repeat one operation, and the repeated operation decides which type a sequence is. The table and the worked examples below show how to tell them apart. Read the difference column and the ratio column side by side, because that comparison is what the classification task asks for.

How to Identify the Type of Sequence

To determine whether a sequence is arithmetic or geometric:

1. Arithmetic Sequence: Calculate the difference between consecutive terms. If the difference is always the same, then the sequence is arithmetic.

   $$\begin{aligned} b &= a_2 - a_1 \\ &= a_3 - a_2 \\ &= a_4 - a_3 \\ &= ... \end{aligned}$$

2. Geometric Sequence: Calculate the ratio between consecutive terms. If the ratio is always the same, then the sequence is geometric.

   $$r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = ...$$

Pattern and Formula Compared

The table places the two types side by side, so the two formulas can be compared row by row.

Worked Classification Examples

The sequence $7,11,15,19,\ldots$ has constant differences:

These equal differences show that the sequence is arithmetic. The sequence $5,10,20,40,\ldots$ satisfies:

Each term is twice the previous term, so the sequence is geometric with $r=2$. The sequence $1,2,4,7,\ldots$ is neither arithmetic nor geometric because its differences $1,2,3$ are not constant and its ratios also vary.

Linear and Exponential Models

A non-constant arithmetic sequence depends linearly on its index. A geometric sequence with a positive ratio other than one depends exponentially on its index. These descriptions identify the mathematical form, but they do not imply that a real process continues unchanged forever.

Adding the same amount each month creates an arithmetic model. Increasing by the same percentage each month creates a geometric model.

Special Cases

The constant zero sequence is arithmetic with $b=0$ and also satisfies $a_{n+1}=ra_n$ for every ratio $r$. A constant non-zero sequence is arithmetic with $b=0$ and geometric with $r=1$.

The arithmetic and geometric categories therefore overlap in these special cases.

Arithmetic and Geometric Models

Each example below has a starting value and one repeated change. The examples name the pattern they follow and the value of $b$ or $r$. Work out both values yourself before you read the labelled answer, then check whether the pattern you found reproduces every term.

Examples of Arithmetic Sequences

Every step adds the same amount, so the difference $b$ stays constant.

1. Regular Savings: A student saves money in the school cooperative with an arithmetic pattern. In the first month, they save $\text{Rp}5{,}000$, in the second month $\text{Rp}7{,}000$, in the third month $\text{Rp}9{,}000$, and so on. With a difference of $\text{Rp}2{,}000$, the amount deposited in month $10$ can be calculated using the arithmetic sequence formula.

2. Plant Growth: In a simplified model, a plant grows $3 \text{ cm}$ taller every week from an initial height of $15 \text{ cm}$. Its modeled height follows an arithmetic sequence.

Examples of Geometric Sequences

Every step multiplies by the same factor, so the ratio $r$ stays constant.

1. Investment with Compound Interest: Suppose $\text{Rp}1{,}000{,}000$ earns $10\%$ interest compounded annually, with no deposits or withdrawals. Its value at the end of each year forms a geometric sequence with a ratio of $1.1$.

2. Idealized Population Growth: A model in which a bacterial population doubles every hour forms a geometric sequence with a ratio of $2$. Real populations eventually face limits that this simple model omits.

Exercise

Classify each sequence and determine $b$ or $r$ when possible. Test the differences first, and switch to quotients only when the differences are not constant. A sequence that fits neither pattern still needs an answer.

1. $18,12,6,0,\ldots$
2. $81,-27,9,-3,\ldots$
3. $2,4,8,14,\ldots$

Solution

Testing the differences and then the ratios gives the classification for each sequence.

1. The first sequence is arithmetic with $b=-6$.
2. The second sequence is geometric with $r=-1/3$.
3. The third sequence is neither arithmetic nor geometric because neither its differences nor its ratios are constant.