Difference between Arithmetic and Geometric Sequence: Sequence and Series - Mathematics (Grade 10)

Understanding Sequences

A number sequence is an arrangement of numbers that follows a specific pattern. Two common types are arithmetic sequences and geometric sequences.

Arithmetic Sequences

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:

a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = ... = a_n - a_{n-1} = b

where $b$ is the constant difference.

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:

U_n = a + (n-1)b

Geometric Sequences

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:

\frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = ... = \frac{a_n}{a_{n-1}} = r

where $r$ is the constant ratio.

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:

U_n = a \times r^{n-1}

Key Differences

How to Identify the Type of Sequence

To determine whether a sequence is arithmetic or geometric:

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

$b = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = ...$
Geometric Sequence: Calculate the ratio between consecutive terms. If the ratio is always the same, then the sequence is geometric.

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

Comparison Table

Aspect	Arithmetic Sequence	Geometric Sequence
Pattern	Constant difference	Constant ratio
Formula for the general term	$U_n = a + (n-1)b$	$U_n = a \times r^{n-1}$
Growth	Linear	Exponential

Applications in Daily Life

Examples of Arithmetic Sequences

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 total savings in month $10$ can be calculated using the arithmetic sequence formula.
Plant Growth: The height of a plant that increases at a constant rate each week. If a plant grows $3 \text{ cm}$ taller every week with an initial height of $15 \text{ cm}$ , then its height follows an arithmetic sequence.

Examples of Geometric Sequences

Investment with Compound Interest: If $\text{Rp}1{,}000{,}000$ is invested with a $10\%$ annual interest rate, the investment value will form a geometric sequence with a ratio of $1.1$ .
Population Growth: Bacteria that double in population every hour form a geometric sequence with a ratio of $2$ .