Sequences

Types of Mathematical Sequences

Now that we understand the basics, let's explore different types of sequences. Each type follows its own unique pattern, and recognizing these patterns is key to working with sequences effectively.

1. Arithmetic Sequences

Definition: A sequence where we add the same number (called the common difference) to get from one term to the next.

General Pattern: $a_{n+1} = a_n + d$ (where $d$ is the common difference)

Example: $3, 7, 11, 15, 19, \ldots$

• $a_1 = 3$
• Common difference $d = 4$
• Pattern: $a_2 = 3 + 4 = 7$, $a_3 = 7 + 4 = 11$, etc.
• General formula: $a_n = 3 + (n-1) \times 4$

2. Geometric Sequences

Definition: A sequence where we multiply by the same number (called the common ratio) to get from one term to the next.

General Pattern: $a_{n+1} = a_n \times r$ (where $r$ is the common ratio)

Example: $2, 6, 18, 54, 162, \ldots$

• $a_1 = 2$
• Common ratio $r = 3$
• Pattern: $a_2 = 2 \times 3 = 6$, $a_3 = 6 \times 3 = 18$, etc.
• General formula: $a_n = 2 \times 3^{n-1}$

3. The Famous Fibonacci Sequence

Definition: Each term is the sum of the two preceding terms.

Pattern: $a_n = a_{n-1} + a_{n-2}$ (starting with $a_1 = 1$, $a_2 = 1$)

The Sequence:$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots$

How it works:

• $a_1 = 1$, $a_2 = 1$ (given starting values)
• $a_3 = a_1 + a_2 = 1 + 1 = 2$
• $a_4 = a_2 + a_3 = 1 + 2 = 3$
• $a_5 = a_3 + a_4 = 2 + 3 = 5$
• $a_6 = a_4 + a_5 = 3 + 5 = 8$

Fun Fact: The Fibonacci sequence appears frequently in nature, from flower petals to spiral galaxies!

4. Quadratic Sequences

Definition: Sequences where the second differences between terms are constant.

Example:$1, 4, 9, 16, 25, \ldots$ (perfect squares)

• First differences: $3, 5, 7, 9, \ldots$
• Second differences: $2, 2, 2, \ldots$ (constant!)
• General formula: $a_n = n^2$

5. Mixed Operation Sequences

Some sequences use combinations of operations:

Example 1: $1, 3, 9, 19, 33, \ldots$

• Pattern: multiply by $2$, then add $1$
• $a_1 = 1$
• $a_2 = (1 \times 2) + 1 = 3$
• $a_3 = (3 \times 2) + 3 = 9$
• $a_4 = (9 \times 2) + 1 = 19$

Example 2: $10, 5, 15, 7.5, 22.5, \ldots$

• Pattern alternates: $\div 2$, then $\times 3$
• $10 \div 2 = 5 \rightarrow 5 \times 3 = 15 \rightarrow 15 \div 2 = 7.5 \rightarrow 7.5 \times 3 = 22.5$

How to Identify Sequence Types

Step-by-step approach:

1. Look at differences between consecutive terms
   • If differences are constant $\rightarrow$ Arithmetic sequence

1. Look at ratios between consecutive terms
   • If ratios are constant $\rightarrow$ Geometric sequence

1. Check second differences
   • If second differences are constant $\rightarrow$ Quadratic sequence

1. Look for recursive patterns
   • Does each term depend on previous terms? $\rightarrow$ Recursive sequence (like Fibonacci)

1. Test for mixed operations
   • Try different combinations of operations

Why Understanding Sequence Types Matters

• Problem Solving: Different types require different solution strategies
• Pattern Recognition: Helps predict future terms
• Real-World Applications: Sequences model everything from population growth to financial investments
• Mathematical Foundation: Essential for calculus, number theory, and advanced mathematics

Next, we will present some more complex sequences of numbers, including one of the most famous: the Fibonacci sequence.

If you are interested in this article, you may also be interested in the following articles:

• The real line
• Opposite numbers
• Integers
• Absolute value

You can find a variety of useful articles about mathematics in Tutorela !

$2, 4, 8, 16, 32$
In this sequence, to get from one term to the next we will multiply by $2$.

$2$

$4=2\times2$

$8=2\times4$

$16=2\times8$

And so on.

$3, 9, 27, 81, 243$
In this sequence, to get from one term to the next we need to multiply by $3$.
$3$

$9=3\times3$

$27=9\times3$

$81=27\times3$

$243=81\times3$

And so on.

$6, 4, 2, 0, -2$

In this sequence, to get from one term to the next we need to subtract $2$.

$6$

$4=6-2$

$2=4-2$

$0=2-2$

$-2=0-2$

$1000, 500, 250, 125, 62.5$
In this example, the operation used is division. In order to get from one term to the next, we divide the number by $2$.

$1000$

$500=1000:2$

$250=500:2$

$125=250:2$

$62.6=125:2$

$320, 80, 20, 5$

The rule of this sequence is to divide each number by $4$ to find the next number.

$320$

$80=320:4$

$20=80:4$

$5=20:4$

Try to work out the rule for each sequence:

• $1,3.75,6.5,9.25,12$
• $7,49,343,2401,16807$
• $0,-15,-30,-45,-60,-75$
• $891,297,99,33,11$
• $2,8,512,134217728$

Sequences are ordered sets of numbers that follow a rule or pattern.

A sequence is a set of ordered numbers. The numbers follow a rule that tells us how to obtain the numbers of the sequence using the previous ones. Many times the rules are governed by the operations of addition, subtraction, multiplication, division, or some combination thereof.

There are many types of sequences. For example, increasing and decreasing sequences, in which the numbers are either increasing or decreasing and following a certain pattern. There are also very famous sequences that have their own name, such as the Fibonacci sequence. In this series, the two previous numbers must be added to obtain the next number.