Generating Sequence Terms

A sequence is a list of terms together with a rule for generating them. In mathematics for university admission, you are expected to handle two primary types of rules: term-to-term rules and position-to-term rules. Understanding how to navigate between these is key to identifying specific values in a progression or determining if a given number is a member of a set.

Term-to-term Rules

A term-to-term rule specifies how to calculate the next term in a sequence based on the current term. These rules require a starting value, usually denoted as $t_1$. For example, a sequence starting with 3 and using a term-to-term rule of adding 4 would be 3, 7, 11, 15, 19, and so on.

Formal notation often uses $t_n$ for the $n$th term and $t_{n+1}$ for the following term. A rule such as $t_1 = 7$ and $t_{n+1} = t_n - 2$ generates the sequence 7, 5, 3, 1, -1. This is also known as a recursive definition.

Worked Example: Using a Term-to-term Rule

Find the next four terms in the sequence defined by $t_1 = 3$ and $t_{n+1} = 2t_n - 1$.

1. For $n = 1$: $t_2 = (2 \times t_1) - 1 = (2 \times 3) - 1 = 5$.
2. For $n = 2$: $t_3 = (2 \times t_2) - 1 = (2 \times 5) - 1 = 9$.
3. For $n = 3$: $t_4 = (2 \times t_3) - 1 = (2 \times 9) - 1 = 17$.
4. For $n = 4$: $t_5 = (2 \times t_4) - 1 = (2 \times 17) - 1 = 33$.

The resulting terms are 5, 9, 17, and 33.

Position-to-term Rules

A position-to-term rule, often referred to as the $n$th term rule, allows you to calculate the value of a term based solely on its position $n$ in the list. This is highly efficient for finding terms far along in a sequence without calculating all intermediate values.

For example, if the rule is $2n - 1$:

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

Worked Example: Finding Terms from an $n$th Term Rule

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

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

Checking Membership and Position

Exam questions often require you to determine if a specific value belongs to a sequence or to find the position index $n$ for a given value. This typically involves solving an algebraic equation.

Worked Example: Deciding if a Number is in a Sequence

Is 272 in the sequence 2, 6, 10, 14, ...? Explain your answer.

The first term is 2 and the term-to-term rule is adding 4. If a term is in this sequence, subtracting the first term (2) must result in a multiple of the common difference (4).

$272 - 2 = 270$.

Since 270 is not a multiple of 4 ($270 \div 4 = 67.5$), 272 is not a term in this sequence. This can also be seen by finding the $n$th term rule, $4n - 2$, and setting it equal to 272, which yields $n = 68.5$, which is not an integer.

Worked Example: Finding the Position of a Term

In the sequence $n^2 + 2n$, which term is 35?

To find the position, set the $n$th term rule equal to 35 and solve for $n$:

$n^2 + 2n = 35$ $n^2 + 2n - 35 = 0$ $(n + 7)(n - 5) = 0$

This gives $n = -7$ or $n = 5$. Since the position $n$ must be a positive integer, we discard the negative root. Therefore, 35 is the 5th term of the sequence.