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Sequence and Series

Difference between Arithmetic and Geometric Sequence

Nabil Akbarazzima Fatih

Mathematics

Understanding Sequences

A number sequence is an arrangement of numbers that follows a specific pattern. And we have already learned about two main types of sequences: 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 a1,a2,a3,...,ana_1, a_2, a_3, ..., a_n, then it is an arithmetic sequence if the difference between consecutive terms is always the same:

a2−a1=a3−a2=a4−a3=...=an−an−1=ba_2 - a_1 = a_3 - a_2 = a_4 - a_3 = ... = a_n - a_{n-1} = b

where bb is the constant difference.

Formula for the nth Term of an Arithmetic Sequence

For an arithmetic sequence with first term aa and common difference bb, the formula for the nth term is:

Un=a+(n−1)bU_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 a1,a2,a3,...,ana_1, a_2, a_3, ..., a_n, then it is a geometric sequence if the ratio between consecutive terms is always the same:

a2a1=a3a2=a4a3=...=anan−1=r\frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = ... = \frac{a_n}{a_{n-1}} = r

where rr is the constant ratio.

Formula for the nth Term of a Geometric Sequence

For a geometric sequence with first term aa and common ratio rr, the formula for the nth term is:

Un=a×rn−1U_n = a \times r^{n-1}

Key Differences

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.

    b=a2−a1=a3−a2=a4−a3=...b = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = ...

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

    r=a2a1=a3a2=a4a3=...r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = ...

Comparison Table

AspectArithmetic SequenceGeometric Sequence
PatternConstant differenceConstant ratio
Formula for the nth termUn=a+(n−1)bU_n = a + (n-1)bUn=a×rn−1U_n = a \times r^{n-1}
GrowthLinearExponential

Applications in Daily Life

Examples of Arithmetic Sequences

  1. Regular Savings: A student saves money in the school cooperative with an arithmetic pattern. In the first month, they save Rp5,000, in the second month Rp7,000, in the third month Rp9,000, and so on. With a difference of Rp2,000, the total savings in the 10th month can be calculated using the arithmetic sequence formula.

  2. Plant Growth: The height of a plant that increases at a constant rate each week. If a plant grows 3 cm taller every week with an initial height of 15 cm, then its height follows an arithmetic sequence.

Examples of Geometric Sequences

  1. Investment with Compound Interest: If Rp1,000,000 is invested with a 10% annual interest rate, the investment value will form a geometric sequence with a ratio of 1.1.

  2. Population Growth: Bacteria that double in population every hour form a geometric sequence with a ratio of 2.

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