Special Sequence Types

To generate a sequence, you use substitution, replacing $n$ in the algebraic expression with positive integers ($1, 2, 3...$). You can also use this formula to check if a specific number belongs to the sequence by setting the formula equal to the number and solving for $n$. If $n$ is a positive whole number, the term exists in the sequence.

Worked Example: Generating Terms and Checking Values

A sequence has the nth term formula $6n - 3$. Generate the first three terms, and determine if 57 is in the sequence.

Step 1: Substitute $n=1, 2, 3$ into the formula to generate terms.

• 1st term ($n=1$): $6(1) - 3 = 3$
• 2nd term ($n=2$): $6(2$

Step 2: Set the formula equal to the target number to check if it is in the sequence.

• $6n - 3 = 57$

Step 3: Solve the linear equation for $n$.

• $6n = 60$
• $n = 10$
• Because 10 is a positive integer, 57 is exactly the 10th term in the sequence.

Arithmetic Progressions (Linear Sequences)

Think of climbing a staircase where every step is exactly the same height. An Arithmetic Progression (often called a linear sequence by OCR) increases or decreases by a constant amount every time. This constant change is called the common difference ($d$). If $d$ is positive, the sequence is increasing; if $d$ is negative, it is decreasing.

Every linear sequence can be written in the form $dn + c$. The $d$ represents your common difference, and the $c$ represents the zeroth term — the theoretical value that would exist at position $n=0$ (found by subtracting the common difference from the first term).

Worked Example: Calculating a Linear nth Term

Find the nth term formula for the sequence: 8, 13, 18, 23...

Step 1: Find the common difference ($d$).

• $13 - 8 = 5$
• $18 - 13 = 5$
• The common difference $d = 5$. The formula starts with $$.

Step 2: Find the zeroth term ($c$).

• Subtract the common difference from the first term: $8 - 5 = 3$
• The zeroth term $c = 3$.

Step 3: Combine into the $dn + c$ format.

• $n$th term = $5n + 3$

Special Sequences: Square, Cube, and Triangular Numbers

You can arrange physical objects like pebbles to reveal hidden mathematical structures. Square numbers are formed by multiplying an integer by itself ($n^2$), creating the sequence $1, 4, 9, 16, 25, 36...$. Cube numbers are the result of multiplying an integer by itself twice ($n^3$), generating the sequence .

Triangular numbers represent the number of dots needed to form a perfectly filled equilateral triangle. The sequence is $1, 3, 6, 10, 15, 21...$. You can calculate the $n$th triangular number directly using a specific formula:

$$\text{nth term} = \frac{n(n+1)}{2}$$

Quadratic Sequences

Not all patterns grow by the same amount each time; some accelerate. A Quadratic sequence contains an $n^2$ term and is identified by looking at the differences between terms. The first difference (the gap between consecutive terms) changes, but the second difference (the gap between the first differences) is always constant.

In the general formula $an^2 + bn + c$, the coefficient $a$ is incredibly important: it is always exactly half of the constant second difference. Interestingly, triangular numbers are actually a type of quadratic sequence where the second difference is 1, leading to an $a$ value of $0.5$.

Geometric Progressions

A single bacteria cell dividing repeatedly creates an exploding population. A Geometric Progression (or exponential sequence) is generated by multiplying the previous term by a constant multiplier called the common ratio ($r$). You can find this ratio by dividing any term by the one immediately before it.

For OCR GCSE, simple geometric sequences typically take the form $r^n$, where the common ratio is a rational number greater than zero (e.g., $r = 2$ or $r = 0.5$). For instance, the formula $3^n$ generates the sequence by continuously multiplying by a common ratio of .

Fibonacci-type Sequences

You can create a sequence simply by adding the recent past together. The famous Fibonacci sequence begins $1, 1, 2, 3, 5, 8, 13...$, where every term after the first two is the sum of the two preceding terms.

Any pattern that follows this additive rule is called a Fibonacci-type sequence, even if it starts with completely different numbers or includes negatives (e.g., $-2, 4, 2, 6, 8, 14...$). In exam questions, you may need to work backwards by subtracting a known term from the term after it to find a missing earlier value.

Various Other Sequences: Fractional, Alternating, and Power-based

Sometimes a single sequence is actually two different patterns cleverly disguised as one. Fractional Sequences are structured as a fraction where the numerator pattern and the denominator pattern follow completely independent rules (often linear). You must determine the $n$th term for the top and bottom separately before combining them into a single fraction.

Alternating Sequences flip signs between positive and negative (e.g., $2, -4, 8, -16...$). This is achieved mathematically by multiplying a base sequence by an alternating component like $(-1)^n$. Finally, Power Sequences involve $n$ as an exponent with a constant shift (e.g., ). These often appear in Higher Tier exams as , written using subscript notation such as .