Sequences

A sequence is an ordered list of numbers following a rule. Two types of rule describe sequences:

Term-to-term rule — describes how to find the next term from the previous one.

Example: start at 3, add 4 each time → 3, 7, 11, 15, 19, …

Position-to-term rule (nth term formula) — gives the value of any term directly from its position $n$.

Example: $T(n)=3n+1$ → $T(1)=4$, $T(2)=7$, $T(3)=10$, …

Worked example — the sequence is defined by $T(n)=n^2+2$. Find the first four terms and the 10th term.

$T(1)=3$, $T(2)=6$, $T(3)=11$, $T(4)=18$, $T(10)=102$

Worked example — a sequence has term-to-term rule "multiply by 3". The first term is 2. Write the first 5 terms.

$2,6,18,54,162$

Triangular numbers: $1,3,6,10,15,21,\dots$ — $T(n)=\frac{n(n+1)}{2}$

Square numbers: $1,4,9,16,25,\dots$ — $T(n)=n^2$

Cube numbers: $1,8,27,64,125,\dots$ — $T(n)=n^3$

Fibonacci-type sequences — each term is the sum of the two preceding terms: $1,1,2,3,5,8,13,21,\dots$

Arithmetic progression (AP) — constant difference $d$ between consecutive terms.

$a,a+d,a+2d,a+3d,\dots$ → $T(n)=a+(n-1)d$

Example: $5,8,11,14,\dots$ has $a=5$, $d=3$: $T(n)=3n+2$ ✓

Geometric progression (GP) — constant ratio $r$ between consecutive terms.

$a,ar,a{r}^2,a{r}^3,\dots$ → $T(n)=a{r}^{n-1}$

Example: $2,6,18,54,\dots$ has $r=3$: $T(n)=2\times 3^{n-1}$

(Extra context — Higher: sequences involving surds (e.g. $\sqrt{2},2,2\sqrt{2},4,\dots$) follow the same geometric rule with $a=\sqrt{2}$, $r=\sqrt{2}$.)

For an arithmetic (linear) sequence with common difference $d$ and first term $a$:

$T(n)=dn+(a-d)$

Method:

1. Find the common difference $d$ (difference between consecutive terms).
2. Multiply $d\times n$ to get the "$d$ times table" sequence.
3. Adjust by the constant needed to match the sequence.

Worked example — find the nth term of $7,11,15,19,\dots$

$d=4$. Start with $4n$: $4,8,12,16,\dots$ Need to add 3 each time to match.

$T(n)=4n+3$ ✓ Check: $T(1)=7$; $T(4)=19$ ✓

Worked example — find the nth term of $20,17,14,11,\dots$

$d=-3$. Start with $-3n$: $-3,-6,-9,-12,\dots$ Need to add 23.

$T(n)=-3n+23$ ✓ Check: $T(1)=20$; $T(2)=17$ ✓

Finding a specific term — is 100 in the sequence $3,7,11,15,\dots$? $T(n)=4n-1=100⟹n=\frac{101}{4}=25.25$ — not an integer, so 100 is not in the sequence.

A quadratic sequence has a constant second difference (not a constant first difference).

Identifying quadratic sequences:

$n$	1	2	3	4	5
$T(n)$	3	8	15	24	35
First diff.		5	7	9	11
Second diff.			2	2	2

Constant second difference of 2 → quadratic. The leading coefficient is $\frac{\text{second difference}}{2}=1$, so $T(n)=n^2+\dots$

Method (Higher) — find the nth term:

1. Note the second difference $d_2$. Leading term: $\frac{d_2}{2}{n}^2$.
2. Subtract $\frac{d_2}{2}{n}^2$ from the sequence; find the linear nth term of the remainder.
3. Combine.

Worked example — find the nth term of $3,8,15,24,35,\dots$

Second difference $=2$; leading term $=n^2$. Subtract: $3-1=2$, $8-4=4$, $15-9=6$, $24-16=8$ → remainder is $2n$.

$T(n)=n^2+2n$ ✓ Check: $T(3)=9+6=15$ ✓; $T(5)=25+10=35$ ✓

Geometric progressions model exponential growth or decay. For ratio $r>1$: growth. For $0<r<1$: decay.

Worked example — a bacteria population starts at 500 and doubles every hour. How many bacteria are present after 6 hours?

$T(n)=500\times 2^{n-1}$. After 6 hours ($n=7$ if counting from hour 0): $T=500\times 2^6=500\times 64=32,000$.

Alternatively: $500\times 2^6=32,000$ (multiply by 2 six times). ✓

Which term exceeds 10,000? $500\times 2^{n-1}>10,000⟹2^{n-1}>20$. Since $2^4=16<20$ and $2^5=32>20$, $n-1=5$, so $n=6$ — the 6th term first exceeds 10,000.

(Extra context — the formula for the sum of a GP is $S_n=\frac{a(r^n-1)}{r-1}$; not required for GCSE but useful background.)

1. nth term of a linear sequence — confusing the difference with the constant

The common difference gives the coefficient of $n$, not the full nth term formula. For $5,8,11,\dots$ the nth term is $3n+2$, not $3n$.

2. Geometric sequence — multiplying by a number other than the ratio

If the sequence is $2,6,18,\dots$ the ratio is 3 (multiply each term by 3). Adding 4 (the difference between the first two terms in the AP) would be wrong.

3. Quadratic sequence — using second difference as leading coefficient directly

The leading coefficient is $\frac{\text{second difference}}{2}$, not the second difference itself. Second diff = 6 → leading term $3n^2$, not $6n^2$.

4. Checking whether a value is a term — forgetting $n$ must be a positive integer

$T(n)=k$ gives a valid term only if $n$ is a positive integer. If you get a fractional or negative $n$, the value is not in the sequence.

Mistake	Correction
"nth term of $4,7,10$: $T(n)=3n$"	$T(n)=3n+1$ — check: $T(1)=4$ ✓
"Is 50 in sequence $3n+2$? Yes, because $3n+2=50⟹n=16$"	$n=16$ is a positive integer, so yes ✓ — but must check!
"Second difference is 4, so leading term is $4n^2$"	Leading term is $\frac{4}{2}{n}^2=2n^2$