Sequences

Introduction to Sequences

• A **sequence** is an ordered set of numbers that follow a rule (e.g., 3, 6, 9, 12... rule: add 3).
• Each number is a **term**; its position is given by **n** $(n=1$ is 1st term).
• A **term-to-term rule** tells how to get the next term (e.g., add 4 each time).
• A **position-to-term rule** (nth term formula) lets you find any term directly (e.g., 8n+2).
• To check if a value belongs to a sequence, set the nth term formula equal to the value and solve for n; if n is a whole number, it is a term.

Continuing Sequences

• Use **first differences** to spot patterns: e.g., 4,7,10,13 → differences +3, next term 16.
• If first differences change by a constant amount (second differences constant), the sequence is quadratic.
• If first differences double each time, the next difference is double the last.
• Be familiar with special sequences: **square numbers** (1,4,9,16...), **cube numbers** (1,8,27,64...), **triangular numbers** (1,3,6,10...).

nth Terms of Linear Sequences

• A **linear sequence** (arithmetic) has a constant **common difference** d.
• The nth term formula is **dn + b**, where d is the common difference and b is the term before the 1st term (zero term).
• To find b, continue the sequence backwards by one term (subtract d from the 1st term).
• Example: 5,7,9,11 → $d=2, b=3$ → nth term $= 2n+3$.

Quadratic Sequences

• A **quadratic sequence** has an nth term involving n²; its **second differences** are constant.
• Compare to square numbers $(n^{2}): e.g., 6,9,14,21,30$ → each term is 5 more than $n^{2}$ → nth term $= n^{2}+5$.
• For simple quadratics $an^{2}+b, a =$ ½ $\times (second difference)$.
• Example: sequence 3,9,19,33,51 → second difference $= 4$ → $a=2$, then find b by comparing to $2n^{2}$.

Cubic Sequences

• A **cubic sequence** has an nth term involving n³; its **third differences** are constant.
• Compare to cube numbers $(n^{3}): e.g., 2,9,28,65,126$ → each term is 1 more than $n^{3}$ → nth term $= n^{3}+1$.
• For simple cubics $an^{3}+b, a =$ ⅙ $\times (third difference)$.
• Know cube numbers up to $5^{3}=125$ and $10^{3}=1000$.

Finding the nth Term: Worked Examples

• **Linear**: Find nth term of -7,-3,1,5,9 → $d=4, b=-11$ → nth term $= 4n-11$.
• **Quadratic**: Find nth term of 6,9,14,21,30 → second $diff=2$ → $a=1$ → compare to $n^{2}$ → nth term $= n^{2}+5$.
• **Cubic**: Sequence 4,25,82,193,376 → third $diff=18$ → $a=3$ → compare to $3n^{3}$ → nth term $= 3n^{3}+1$.
• Always check your formula by substituting $n=1,2,3$ to see if it matches the given terms.

Using nth Term to Find Terms and Check Membership

• To find a specific term, substitute n into the formula (e.g., 20th term of $n^{2}+5 = 400+5=405)$.
• To check if a number is in the sequence, set formula = number and solve for n; if n is a positive integer, it is a term.
• Example: Is 98 in sequence 8n+2? $8n+2=98$ → $n=12$ → yes, 12th term.
• If n is not an integer $(e.g., n=15.25)$, the number is not in the sequence.

Pattern Sequences (Exam Style)

• Many exam questions involve patterns of counters, lines, or squares; find the nth term for each quantity.
• Example: Pattern n has black counters $= 2n+2$, white counters $= n^{2}$.
• Use the nth term to check if a pattern can be made with given resources (e.g., 30 black, 140 white for Pattern 12).
• For linear patterns, the number of rows or houses can be found using the nth term formula.

Linear Sequence: Common Difference

Quadratic Sequence: Second Differences

Cubic Sequence: Third Differences

Pattern Sequence: Counters Example