Sequences

Introduction to Sequences

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

nth Terms of Linear Sequences

• A **linear (arithmetic) sequence** has a constant **common difference (d)** between consecutive terms.
• The nth term formula is **dn + b**, where d is the common difference and b is the term before the first term (zero term).
• To find b, continue the sequence backwards by one term (subtract d from the first term).
• Example: For 5, 7, 9, 11… $d=2, b=3$ → nth term $= 2n+3$.
• For decreasing sequences, d is negative (e.g., 15,10,5… $d=-5, b=20$ → nth term $= -5n+20)$.

Quadratic Sequences

• A **quadratic sequence** has an nth term involving n²; its **second differences** are constant.
• To find the nth term: Step 1 – find first and second differences. Step 2 – halve the second difference to get **a** (coefficient of $n^{2})$.
• Step 3 – write out $an^{2}$ and subtract from the original sequence to get a linear sequence. Step 4 – find the nth term of that linear sequence (bn+c).
• Step 5 – add $an^{2} + bn + c$ to get the nth term formula.
• Example: For 6,9,14,21,30… second $difference=2$ → a=1; an²=1,4,9,16,25; $difference=5,5,5,5,5$ → linear nth term=5; so nth term $= n^{2}+5$.
• Common simple quadratics: $n^{2}+1, 2n^{2}, (n+1)^{2}-2$, etc.

Other Sequences

• **Cubic sequences** have constant third differences; nth term involves $n^{3}$. The coefficient $a = (third difference)/6$.
• **Exponential (geometric) sequences** multiply by a constant **common ratio (r)** each time; nth term $= a \times$ rⁿ $(or a \times$ rⁿ⁻¹).
• Common sequences: prime numbers (2,3,5,7…), triangular numbers (1,3,6,10…), cube numbers (1,8,27,64…).
• Sequences can be combined: e.g., fractions with linear numerator and cubic denominator, or sums of two sequences.
• Harder problems may involve setting up and solving equations (e.g., simultaneous equations) to find unknown terms or ratios.

Fibonacci and Special Sequences

• In a **Fibonacci sequence**, each term is the sum of the two previous terms (e.g., 1,1,2,3,5,8…).
• Given first two terms a and b, the 3rd term is a+b, 4th is a+2b, 5th is 2a+3b, 6th is 3a+5b.
• To find unknown terms in a Fibonacci sequence, set up equations using given terms and solve.
• Some sequences follow a pattern like adding a decreasing amount or using a specific rule (e.g., term-to-term rules).

Finding Terms and Checking Membership

• To find a specific term, substitute n into the nth term formula (e.g., 20th term of $n^{2}+5 = 20^{2}+5 = 405)$.
• To check if a number is in a sequence, set the nth term equal to the number and solve for n; if n is a positive integer, it is a term.
• Example: Is 98 in the sequence 8n+2? $8n+2=98$ → $n=12$, yes. Is 124? $8n+2=124$ → $n=15.25$, no.
• Always write the position number above each term in the exam to help spot patterns.

Linear Sequence: Finding nth Term

Quadratic Sequence: Method of Differences

Exponential Sequence: Common Ratio

Fibonacci Sequence: Term Relationship