Sequences: patterns, nth term, recurrence relations

Sequences – Patterns, the $n^{\text{th}}$ Term and Recurrence Relations

1. What is a sequence?

A sequence is an ordered list of numbers written in a definite pattern. Each number is called a term and its position is indicated by a subscript: $a_1, a_2, a_3,\dots$.

2. How to recognise the pattern

Before writing a formula for $a_n$ you must identify the rule that generates the sequence. The table below gives the quickest way to match a pattern to its type – a useful “exam‑style” checklist.

Observed regularity	Sequence type	What to check
Constant first difference $d$	Linear (arithmetic)	$a_{n+1}-a_n = d$ for all $n$
Constant ratio $r$	Geometric (direct proportion)	$\dfrac{a_{n+1}}{a_n}=r$ for all $n$
Constant second difference $2a$	Quadratic (simple quadratic)	First differences increase by the same amount each step
Constant third difference $6a$	Simple cubic (C2.7 requirement)	Second differences increase by a constant amount
Alternating signs, repeating blocks, or “each term = sum of previous terms”	Extended patterns (Fibonacci‑type, alternating, etc.)	Look for a recurrence relation

3. Core content (C2.7) – linear, quadratic and simple cubic sequences

3.1 Linear (arithmetic) sequences

• Pattern: constant first difference $d$.
• Explicit formula:
  \[ a_n = a_1 + (n-1)d \]
• Finding $a$, $b$, $c$ is not required – only $a_1$ and $d$.

Example (Core)

Sequence $7,12,17,\dots$

• $a_1=7$, $d=12-7=5$
• $a_{10}=7+(10-1)\times5=52$

3.2 Quadratic (simple quadratic) sequences

• Pattern: constant second difference $2a$.
• General form: \[ a_n = an^{2}+bn+c \]
• Method to determine $a,b,c$:
1. Calculate the second difference; set $2a=$ that constant.
2. Substitute $n=1,2,3$ (or any three known terms) into the formula to obtain three linear equations.
3. Solve the system for $a,b,c$.

Example (Core)

Sequence $2,6,12,20,30,\dots$

• First differences: $4,6,8,10$ → second difference $2$ (constant).
• $2a = 2 \;\Rightarrow\; a=1$.
• Using $n=1$ and $n=2$: \[ \begin{cases} 1+b+c = 2\\[2mm] 4+2b+c = 6 \end{cases} \;\Longrightarrow\; b=1,\;c=0. \end{cases} \]
• Hence $a_n = n^{2}+n$.

3.3 Simple cubic sequences (C2.7)

• Pattern: constant third difference $6a$.
• General form: \[ a_n = an^{3}+bn^{2}+cn+d \]
• Method to determine $a,b,c,d$:
1. Compute the third differences; set $6a$ equal to that constant.
2. Substitute $n=1,2,3,4$ (four known terms) into the formula to obtain four equations.
3. Solve the system for $a,b,c,d$.

Worked cubic example (Core)

Sequence $1,8,27,64,125,\dots$ (the cubes of the natural numbers).

1. First differences: $7,19,37,61$
2. Second differences: $12,18,24$
3. Third differences: $6,6$ (constant $\Rightarrow a=1$ because $6a=6$).
4. Assume $a_n = n^{3}+bn^{2}+cn+d$ and use the first four terms:
   • $n=1$: $1+b+c+d = 1$
   • $n=2$: $8+4b+2c+d = 8$
   • $n=3$: $27+9b+3c+d = 27$
   • $n=4$: $64+16b+4c+d = 64$
   Subtracting each equation from the next gives $b=c=d=0$.
5. Result: $a_n = n^{3}$, which indeed generates $1,8,27,64,125,\dots$.

4. Extended content (E2.7) – geometric, Fibonacci‑type and other patterns

4.1 Geometric sequences (direct proportion)

• Pattern: constant ratio $r$.
• Explicit formula: \[ a_n = a_1\,r^{\,n-1} \]
• Link to the syllabus: geometric sequences are an application of the “direct proportion” sub‑topic (E2.8).

Example (Extended)

Sequence $3,9,27,81,\dots$

• $a_1=3$, $r=9/3=3$
• $a_n = 3\cdot3^{\,n-1}=3^{\,n}$

4.2 Fibonacci‑type (and other recurrence) sequences

• Pattern: each term is expressed as a sum (or combination) of earlier terms.
• Typical recurrence: \[ a_n = a_{n-1}+a_{n-2}\qquad(n\ge3) \]
• Only the first two terms need to be given.

Example (Extended)

Sequence $5,8,13,21,34,\dots$

\[ a_1=5,\; a_2=8,\; a_n = a_{n-1}+a_{n-2}\;(n\ge3). \]

4.3 Converting between explicit and recursive forms

1. Identify the pattern (difference, ratio, or recurrence).
2. Derive the explicit formula using the appropriate general form.
3. Write the corresponding recurrence by expressing $a_n$ in terms of earlier terms (e.g., $a_n=a_{n-1}+d$ for arithmetic).
4. Check consistency by substituting a few values of $n$.

Conversion example (Extended)

Arithmetic sequence $a_n = 4n-3$.

• Recurrence: $a_n = a_{n-1}+4$, with $a_1 = 1$.

5. Solving equations that arise from sequences

When a question asks “find $n$ such that $a_n = k$”, the explicit formula often leads to a quadratic or cubic equation.

5.1 Quadratic example

For the quadratic sequence $a_n = n^{2}+n$, find $n$ when $a_n = 30$.

\[ n^{2}+n-30=0\quad\Longrightarrow\quad n=\frac{-1\pm\sqrt{1+120}}{2} =\frac{-1\pm11}{2}. \] Only the positive integer solution is accepted: $n=5$ (since $5^{2}+5=30$).

5.2 Cubic example (Extended)

For the cubic sequence $a_n = n^{3}+n^{2}+n$, determine $n$ when $a_n = 120$.

\[ n^{3}+n^{2}+n-120=0. \] Testing integer values (or using the Rational Root Theorem) shows $n=4$ satisfies the equation: \[ 4^{3}+4^{2}+4=64+16+4=84eq120,\qquad 5^{3}+5^{2}+5=125+25+5=155eq120. \] Thus there is **no integer solution**; the problem would be rejected in an exam. (If the cubic were $a_n=n^{3}$, then $n^{3}=120$ gives $n\approx4.93$, which is not an integer, so the answer would be “no integer $n$”.) The point is to demonstrate that students must be comfortable solving the resulting quadratic or cubic and interpreting the result in the context of the sequence.

6. Recurrence (recursive) relations – a quick reference

Sequence type	Recurrence relation	Initial conditions required
Arithmetic	$a_n = a_{n-1}+d$	$a_1$
Geometric	$a_n = r\,a_{n-1}$	$a_1$
Quadratic (simple)	$a_n = 2a_{n-1}-a_{n-2}+2a$ (derived from constant second difference)	$a_1,a_2$
Simple cubic	$a_n = 3a_{n-1}-3a_{n-2}+a_{n-3}$ (holds for $a_n=n^{3}$)	$a_1,a_2,a_3$
Fibonacci‑type	$a_n = a_{n-1}+a_{n-2}$	$a_1,a_2$

7. Common pitfalls and tips

• Never assume a sequence is arithmetic or geometric without checking the first difference or ratio.
• For quadratic and cubic sequences verify that the second or third differences are truly constant before using the polynomial form.
• A recurrence relation must be supplied with enough initial terms (one for arithmetic/geometric, two for Fibonacci‑type, three for cubic) to generate a unique sequence.
• Always test your explicit formula with at least two given terms; if it fails, re‑examine the pattern.
• When solving for coefficients in quadratic or cubic formulas, work systematically (substitute $n=1,2,3,\dots$) to avoid arithmetic errors.
• Remember that geometric sequences are an application of direct proportion (E2.8); the constant ratio $r$ is the constant of proportionality.

8. Summary checklist (exam‑style)

Core (C2.7) – required for all candidates

1. Identify whether the first, second or third differences are constant.
2. Choose the correct explicit form:
   • Linear → $a_n = a_1+(n-1)d$
   • Quadratic → $a_n = an^{2}+bn+c$
   • Simple cubic → $a_n = an^{3}+bn^{2}+cn+d$
3. Determine the unknown coefficients using the minimum number of given terms (2 for linear, 3 for quadratic, 4 for cubic).
4. If the question asks for a specific term, substitute the required $n$.
5. If the question asks for $n$ given a term value, solve the resulting quadratic or cubic equation and interpret the solution.

Extended (E2.7) – additional material

6. Check for a constant ratio → use the geometric formula $a_n = a_1 r^{\,n-1}$.
7. Look for a recurrence pattern (e.g., Fibonacci‑type) → write the appropriate relation and note the required initial terms.
8. Convert between explicit and recursive forms when asked.
9. Recall the link to direct proportion (E2.8) for geometric sequences.