Generating Sequence Terms using Rules

What is a Sequence?

A sequence is defined as a list of terms accompanied by a rule for generating them. In Mathematics 1 for the ESAT, you will encounter two primary types of rules: term-to-term rules and position-to-term rules. Understanding the difference between these is vital for correctly identifying and generating the elements of a progression.

Term-to-term Rules

A term-to-term rule indicates how to move from one term in the sequence to the next term. These rules require a starting point, usually the first term, to begin the generation process.

For example, a sequence might be described by its first term and its rule. If the first term is $3$ and the term-to-term rule is $+4$, the terms are $3, 7, 11, 15, 19, \dots$ because you add $4$ each time to find the subsequent value.

Mathematical Notation for Term-to-term Rules

Term-to-term rules are often expressed using specific notation. We use $t_1$ to represent the first term and $t_n$ to represent the $n$th term. Consequently, $t_{n+1}$ represents the term immediately following $t_n$.

Consider the rule: $t_1 = 7$ and $t_{n+1} = t_n - 2$. This tells us to start at $7$ and subtract $2$ to find each following term, resulting in the sequence $7, 5, 3, 1, -1, \dots$.

Generating Sequences Using Term-to-term Rules

To generate multiple terms, apply the rule recursively. Suppose you are asked to find the next $4$ terms in this sequence: $t_1 = 3$ and $t_{n+1} = 2t_n - 1$.

1. The first term is given: $t_1 = 3$.
2. The second term: $t_2 = (2 \times 3) - 1 = 5$.
3. The third term: $t_3 = (2 \times 5) - 1 = 9$.
4. The fourth term: $t_4 = (2 \times 9) - 1 = 17$.
5. The fifth term: $t_5 = (2 \times 17) - 1 = 33$.

Position-to-term Rules

A position-to-term rule, or $n$th term rule, describes the relationship between the position of a term in the sequence and the value of the term itself. This allows you to find any specific term, such as the $20$th term, without having to calculate all the terms that come before it.

For example, consider the rule $2n - 1$:

• The $3$rd term is $5$ because $(2 \times 3) - 1 = 5$.
• The $8$th term is $15$ because $(2 \times 8) - 1 = 15$.

Deciding Whether a Number is in a Particular Sequence

You may be asked to determine if a specific number belongs to a given sequence. For arithmetic sequences, you can use the first term and the common difference to test membership.

Example: Is $272$ in the sequence $2, 6, 10, 14, \dots$?

1. Identify the first term, which is $2$, and the term-to-term rule, which is $+4$.
2. If a term belongs to this sequence, subtracting the first term must result in a multiple of the common difference.
3. Test the value: $272 - 2 = 270$.
4. Check divisibility: $270$ is not a multiple of $4$ because $270 \div 4 = 67.5$.

Conclusion: $272$ is not a term in this sequence.

Finding Terms from the $n$th Term Rule

When given a position-to-term rule, you can find any term by substituting the position number $n$ into the formula.

Example: Find the $4$th and $10$th terms for the sequence whose $n$th term rule is $2n + 3$.

• To find the $4$th term, let $n = 4$: $(2 \times 4) + 3 = 11$.
• To find the $10$th term, let $n = 10$: $(2 \times 10) + 3 = 23$.

Finding the Position of a Particular Term

If you know a value is in a sequence, you can determine its position by setting the $n$th term rule equal to that value and solving for $n$.

Example: In the sequence defined by $n^2 + 2n$, which term has the value $35$?

1. Create an equation: $n^2 + 2n = 35$.
2. Rearrange to solve: $n^2 + 2n - 35 = 0$.
3. Factorise: $(n + 7)(n - 5) = 0$.
4. Possible values for $n$ are $-7$ or $5$.
5. Since the position $n$ must be a positive integer, we conclude $n = 5$.

Therefore, $35$ is the $5$th term of the sequence.