Sequences

Spec reference: Algebra - Sequences
Key idea: Find terms and nth term rules for arithmetic and geometric sequences.

---

---

Arithmetic sequences

An arithmetic sequence increases or decreases by a fixed amount each term - the common difference $d$.

Example

$5,8,11,14,17,\dots$

Common difference: $d=3$

---

nth term of an arithmetic sequence

$$T_n=a+(n-1)d$$

where $a$ is the first term and $d$ is the common difference.

Or simply: find the term-to-term rule, then work backwards to the "zeroth term".

Example

Find the nth term of $3,7,11,15,\dots$

$d=4$, so the nth term involves $4n$.

When $n=1$: $4(1)=4$, but the first term is 3, so subtract 1.

$$T_n=\boxed{{\displaystyle 4n-1}}$$

Check: $n=1$: $4-1=3$ ✓, $n=2$: $8-1=7$ ✓

---

Using the nth term

Example

Is 75 a term in the sequence $4n-1$?

Set $4n-1=75$

$4n=76$, so $n=19$

Yes - it is the 19th term.

Example

Is 50 a term in the sequence $4n-1$?

$4n-1=50$ → $4n=51$ → $n=12.75$

Not a whole number, so no, 50 is not in the sequence.

---

Geometric sequences

A geometric sequence multiplies by a fixed common ratio $r$ each term.

Example

$2,6,18,54,\dots$

Common ratio: $r=3$

---

Special sequences

Sequence	nth term
Square numbers: $1,4,9,16,\dots$	$n^2$
Cube numbers: $1,8,27,64,\dots$	$n^3$
Triangular numbers: $1,3,6,10,\dots$	$\frac{n(n+1)}{2}$
Fibonacci: $1,1,2,3,5,8,\dots$	Each term = sum of previous two

---

Quadratic sequences

If the second differences are constant, the sequence is quadratic and the nth term includes $n^2$.

Example

Find the nth term of $3,8,15,24,35,\dots$

First differences: $5,7,9,11$ (not constant)

Second differences: $2,2,2$ (constant = 2)

Start with $n^2$: $1,4,9,16,25$

Difference from sequence: $2,4,6,8,10$ → this is $2n$

$$T_n=\boxed{{\displaystyle n^2+2n}}$$

---

Exam tips

Watch out for

• Always verify your nth term formula with at least two terms
• A common difference of 0 means all terms are the same
• Geometric sequences can involve fractions or decimals as the ratio

---

Test Yourself

Question 1 of 5

nth term of 5, 8, 11, 14, ...?