Generating Sequences and Linear Rules

Step 1: Find the constant difference between consecutive terms.

• $11 - 7 = 4$ and $15 - 11 = 4$. The common difference is $+4$.
• The term-to-term rule is "add $4$".

Step 2: Apply the rule to find the next three terms.

• Term 5: $19 + 4 = 23$
• Term 6: $23 + 4 = 27$
• Term 7: $27 + 4 = 31$
• The next three terms are $23, 27, 31$.

Worked Example: Generating from a Rule

The first term of a sequence is $24$. The term-to-term rule is "subtract $5$". Calculate the first five terms.

Step 1: State the first term.

• Term 1 = $24$

Step 2: Subtract $5$ repeatedly for each subsequent term.

• Term 2 = $24 - 5 = 19$
• Term 3 = $19 - 5 = 14$
• Term 4 = $14 - 5 = 9$
• Term 5 = $9 - 5 = 4$
• The first five terms are $24, 19, 14, 9, 4$.

Finding the nth Term of an Arithmetic Sequence

Finding the 100th term of a sequence by repeatedly adding the common difference would take ages, but an algebraic formula calculates it instantly. This formula is called the position-to-term rule, or the $n^{th}$ term formula, because it links the term's value directly to its position number ($n$).

• For an arithmetic sequence, the $n^{th}$ term is written in the form $an + b$.
• The letter $a$ represents the common difference.
• The letter $b$ is the constant "zero term" — the value the sequence would have at position $n = 0$ (one step backwards from the first term).

Worked Example: Finding the Position-to-Term Rule

Determine the $n^{th}$ term formula for the arithmetic sequence $6, 13, 20, 27 \dots$

Step 1: Calculate the common difference ($a$).

• $13 - 6 = 7$. The sequence increases by $7$ each time, so $a = 7$.
• The formula begins with $7n$.

Step 2: Find the constant zero term ($b$) using the first term.

• The first term (where $n = 1$) is $6$.
• Move backwards one step by subtracting the common difference: $6 - 7 = -1$.
• Therefore, $b = -1$.

Step 3: Write the final expression.

• The $n^{th}$ term is $7n - 1$.

Generating Sequences Using Substitution

If a formula gives you an output based on the input number you provide, how do you find the first five terms? You use substitution, replacing the variable $n$ with positive integers ($n = 1, 2, 3 \dots$) corresponding to the position in the sequence. OCR exam papers often use subscript notation, such as $u_n$, to represent the $n^{th}$ term.

Worked Example: Substituting into a Formula

A sequence is generated by the formula $u_n = 4n - 3$. Calculate the first five terms of the sequence.

Step 1: Substitute $n = 1$ for the first term.

• $u_1 = 4(1) - 3 = 4 - 3 = 1$

Step 2: Substitute $n = 2$ for the second term.

• $u_2 = 4(2) - 3 = 8 - 3 = 5$

Step 3: Substitute $n = 3, 4, \text{and } 5$ for the remaining terms.

• $u_3 = 4(3) - 3 = 12 - 3 = 9$
• $u_4 = 4(4) - 3 = 16 - 3 = 13$
• $u_5 = 4(5) - 3 = 20 - 3 = 17$
• The first five terms are $1, 5, 9, 13, 17$.

Checking if a Number is in a Sequence

• To verify if a specific number belongs in a sequence, set the $n^{th}$ term formula equal to that number and solve for $n$.
• Because $n$ represents a position, it must be a positive integer (whole number).
• If solving the equation results in a decimal or fraction, the number is not in the sequence.

Worked Example: Verifying a Term

Is $85$ in the sequence given by the $n^{th}$ term $6n + 2$?

Step 1: Set up the equation.

• $6n + 2 = 85$

Step 2: Solve for $n$.

• $6n = 85 - 2$
• $6n = 83$
• $n = 83 \div 6 = 13.833\dots$

Step 3: State your conclusion.

• No, $85$ is not in the sequence because $n$ is not a positive integer.