Every time you stack a new layer of bricks to build a pyramid, the number of bricks changes by a predictable amount. In mathematics, an ordered set of numbers that follows a specific pattern is called a sequence, and each individual number is called a term.
Identify the term-to-term rule for the sequence and calculate the next three terms.
Step 1: Find the constant difference between consecutive terms.
Step 2: Apply the rule to find the next three terms.
The first term of a sequence is . The term-to-term rule is "subtract ". Calculate the first five terms.
Step 1: State the first term.
Step 2: Subtract repeatedly for each subsequent term.
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 term formula, because it links the term's value directly to its position number ().
Determine the term formula for the arithmetic sequence
Step 1: Calculate the common difference ().
Step 2: Find the constant zero term () using the first term.
Step 3: Write the final expression.
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 with positive integers () corresponding to the position in the sequence. OCR exam papers often use subscript notation, such as , to represent the term.
A sequence is generated by the formula . Calculate the first five terms of the sequence.
Step 1: Substitute for the first term.
Step 2: Substitute for the second term.
Step 3: Substitute for the remaining terms.
Is in the sequence given by the term ?
Step 1: Set up the equation.
Step 2: Solve for .
Step 3: State your conclusion.
Students often forget to include the negative sign in the common difference for descending sequences (e.g., writing instead of ).
Do not confuse a term-to-term rule (like "add 4") with the nth term formula (which must include the letter n, like ). OCR frequently asks you to "Write an expression for the nth term", requiring the algebraic formula.
When finding the nth term, always write the position numbers (1, 2, 3...) directly above the sequence terms to avoid confusing the term's value with its position.
In OCR questions asking "Is this number in the sequence?", you must state clearly in your final conclusion that your solved value for n is (or is not) a positive integer to secure the final communication mark.
Sequence
An ordered set of numbers that follow a specific mathematical rule or pattern.
Term
A single number or element within a sequence.
Term-to-term rule
A rule that defines how to calculate the next term in a sequence using the value of the current term.
Arithmetic progression
A sequence where the difference between any two consecutive terms is a constant value.
Linear sequence
Another name for an arithmetic progression; a sequence that increases or decreases by the same amount each time.
Common difference
The constant amount added to, or subtracted from, each term to get the next term in an arithmetic sequence.
Geometric sequence
A sequence where each term is found by multiplying the previous term by a fixed, non-zero number.
Common ratio
The constant value every term is multiplied by to get the next term in a geometric sequence.
Fibonacci-type sequence
A sequence where each term is generated by adding the two previous terms together (e.g. 1, 1, 2, 3, 5, 8...).
Position-to-term rule
An algebraic formula that allows you to calculate the value of any term directly from its position number (n) in the sequence.
nth term
An algebraic expression, often written as , used to calculate the value of a term based on its position .
Substitution
The mathematical process of replacing a variable (like n) in a formula with a specific numerical value to calculate an output.
Put your knowledge into practice — try past paper questions for Mathematics
Sequence
An ordered set of numbers that follow a specific mathematical rule or pattern.
Term
A single number or element within a sequence.
Term-to-term rule
A rule that defines how to calculate the next term in a sequence using the value of the current term.
Arithmetic progression
A sequence where the difference between any two consecutive terms is a constant value.
Linear sequence
Another name for an arithmetic progression; a sequence that increases or decreases by the same amount each time.
Common difference
The constant amount added to, or subtracted from, each term to get the next term in an arithmetic sequence.
Geometric sequence
A sequence where each term is found by multiplying the previous term by a fixed, non-zero number.
Common ratio
The constant value every term is multiplied by to get the next term in a geometric sequence.
Fibonacci-type sequence
A sequence where each term is generated by adding the two previous terms together (e.g. 1, 1, 2, 3, 5, 8...).
Position-to-term rule
An algebraic formula that allows you to calculate the value of any term directly from its position number (n) in the sequence.
nth term
An algebraic expression, often written as , used to calculate the value of a term based on its position .
Substitution
The mathematical process of replacing a variable (like n) in a formula with a specific numerical value to calculate an output.