Continuing sequences, finding the nth term of linear and quadratic sequences, and recognising geometric, triangular and Fibonacci sequences.

What this dot point is asking

AQA wants you to continue a sequence, find a formula for its $n$th term, and recognise the named special sequences. Linear (arithmetic) sequences appear at both tiers; quadratic sequences and the geometric, triangular and Fibonacci patterns are Higher-tier favourites. The $n$th term turns a list into a rule, letting you find any term without writing out all the ones before it.

Continuing a sequence

To continue a sequence, work out the rule connecting consecutive terms. Most often it is a common difference (add the same amount each time) or a common ratio (multiply by the same amount). For $5, 8, 11, 14, \ldots$ you add $3$, so the next term is $17$. For $2, 6, 18, 54, \ldots$ you multiply by $3$, so the next term is $162$.

The nth term of a linear sequence

For $7, 12, 17, 22, \ldots$ the common difference is $5$, so the $n$th term starts as $5n$. Checking $n = 1$ gives $5$, but the sequence starts at $7$, so add $2$: the $n$th term is $5n + 2$. You can then find the 100th term directly as $5(100) + 2 = 502$, or test whether $158$ is a term by solving $5n + 2 = 158$, giving $n = 31.2$, which is not a whole number, so $158$ is not in the sequence.

Quadratic sequences at Higher tier

A quadratic sequence has a constant second difference and an $n$th term of the form $an^2 + bn + c$.

Special sequences to recognise

• Geometric: a constant ratio between terms, such as $2, 6, 18, 54$ (ratio $3$).
• Triangular numbers: $1, 3, 6, 10, 15, \ldots$, formed by adding $1, 2, 3, \ldots$; the $n$th term is $\tfrac{1}{2}n(n + 1)$.
• Square numbers: $1, 4, 9, 16, \ldots$ with $n$th term $n^2$.
• Cube numbers: $1, 8, 27, 64, \ldots$ with $n$th term $n^3$.
• Fibonacci-type: each term is the sum of the previous two, as in $1, 1, 2, 3, 5, 8, \ldots$ or any starting pair.

Using the nth term to answer questions

Once you have the $n$th term, two standard questions follow. To find a specific term, substitute its position: the $20$th term of $4n + 3$ is $4 \times 20 + 3 = 83$. To test whether a value is in the sequence, set the $n$th term equal to it and solve for $n$; if $n$ is a positive whole number the value is in the sequence, otherwise it is not. For $4n + 3 = 99$, solving gives $n = 24$, a whole number, so $99$ is the $24$th term, but $4n + 3 = 100$ gives $n = 24.25$, so $100$ is not a term. This "is it a term" test is a reliable exam question.

Recognising the type from the differences

The differences are the diagnostic tool for any sequence. Constant first differences mean a linear sequence and a $dn + c$ rule. Constant second differences mean a quadratic sequence and an $an^2 + bn + c$ rule. A constant ratio between terms (rather than a constant difference) signals a geometric sequence. Checking the first differences, and then the second differences if the first are not constant, is the systematic first move that tells you which method to apply before committing to any algebra.