A sequence is an ordered set of numbers (or terms) determined by a pre-defined rule.

Each member of a sequence is called a term of the sequence. A term of a sequence is usually represented by the letter "t". A specific term of a sequence is usually represented by the expression "t_n", where "n" is the term number or position of the term in the sequence.

A sequence can be finite (has a countable number of terms) or infinite (with terms continuing indefinitely).

A finite sequence with six terms could be represented as shown below:

t₁, t₂, t₃, t₄, t₅, t₆

In the sequence above, t₁ symbolically represents the first term of the sequence, t₂ symbolically represents the second term of the sequence, t₃ symbolically represents the third term of the sequence, and so on.

A finite sequence with six-thousand terms could be represented as shown below:

t₁, t₂, t₃, . . ., t₆₀₀₀      or      t₁, t₂, t₃, . . ., t_n, . . ., t₆₀₀₀

An infinite sequence could be represented as shown below:

t₁, t₂, t₃, . . .      or      t₁, t₂, t₃, . . ., t_n, . . .

A sequence without a determining rule is often referred to as a pattern.

A rule can determine each term of a sequence directly (by using an equation), or a rule can determine each term of a sequence recursively (by using preceding terms).

Defining Rule

t_n = 2n + 3

To generate the sequence, substitute the natural numbers into the equation.

Sequence: 5, 7, 9, 11, 13, 15, 17, . . .

Sequence: Starting with five, two is added to each term of the sequence to determine the next term of the sequence.

Defining Rule

t₁ = 5

t_n = t_n-1 + 2

Knowing that the first terms is 5 (t₁ = 5), calculate the value of the second term (t₂).

Since we need to calculate t₂, substitute n = 2 into the second part of the recursive rule (t_n = t_n-1 + 2)

      t₍₂₎ = t_(2)-1 + 2

      t₂ = t₁ + 2                  (But it is known that t₁ = 5)

      t₂ = 5 + 2

      t₂ = 7

Repeat this process to find t₃.

Knowing that the first two terms are 5 and 7 (t₁ = 5 and t₂ = 7), calculate the value of the third term (t₃).

Since we need to calculate t₃, substitute n = 3 into the second part of the recursive rule (t_n = t_n-1 + 2)

      t₍₃₎ = t_(3)-1 + 2

      t₃ = t₂ + 2                  (But it was calculated that t₂ = 7)

      t₃ = 7 + 2

      t₃ = 9

Repeat this process to find t₄.

Knowing that the first three terms are 5, 7 and 9 (t₁ = 5, t₂ = 7 and t₃ = 9), calculate the value of the fourth term (t₄).

Since we need to calculate t₄, substitute n = 4 into the second part of the recursive rule (t_n = t_n-1 + 2)

      t₍₄₎ = t_(4)-1 + 2

      t₄ = t₃ + 2                  (But it was calculated that t₃ = 9)

      t₄ = 9 + 2

      t₄ = 11

Repeat this process to find as many terms as required.

Sequence: 5, 7, 9, 11, 13, 15, 17, . . .

Sequence: Starting with five, two is added to each term of the sequence to determine the next term of the sequence.

Defining Rule (Equation) Example 1: t_n = 5n - 4

Defining Rule (Equation) Example 2: t_n = 25 - 3n

Defining Rule (Equation) Example 3: t_n = 5(2)ⁿ

Defining Rule (Equation) Example 4: t_n = 2(5)ⁿ

Defining Rule (Equation) Example 5: t_n = n²

Defining Rule (Recursion) Example 1

t₁ = 20

t_n = t_n-1 - 10

Defining Rule (Recursion) Example 2

t₁ = 3

t_n = 2t_n-1

Defining Rule (Recursion) Example 3

t₁ = 1

t₂ = 1

t_n = t_n-2 + t_n-1

Defining Rule (Recursion) Example 4

t₁ = 2

t₂ = 3

t₂ = 1

t_n = t_n-3 x t_n-2 x t_n-1