Sequences Quizzes
Visual overview of Sequences.
Introduction
A sequence is an ordered list of numbers that follow a rule. In GCSE Maths, recognising the rule lets you predict future terms, find missing terms, and describe patterns algebraically. Sequences appear in arithmetic and geometric forms, as well as special patterns (squares, cubes, triangular numbers). They’re essential for algebra, problem solving, and real-life contexts such as finance, growth, and design.
Example: \(2,4,6,8,\dots\) is arithmetic with common difference \(d=2\).
Core Concepts
Types of Sequences
- Arithmetic: add/subtract a constant difference each step.
- Geometric: multiply/divide by a constant ratio each step.
- Fibonacci-type: each term is the sum of the previous two.
- Special: squares, cubes, triangular numbers, etc.
Arithmetic Sequences
General form: \(a,\,a+d,\,a+2d,\,a+3d,\dots\)
- Example: \(3,7,11,15,\dots\) has \(d=4\).
- n-th term: \(T_n=a+(n-1)d\).
- Example: \(T_5=3+4\times4=19\).
Geometric Sequences
General form: \(a,\,ar,\,ar^2,\,ar^3,\dots\)
- Example: \(2,6,18,54,\dots\) has ratio \(r=3\).
- n-th term: \(T_n=a\cdot r^{\,n-1}\).
- Example: \(T_4=2\cdot3^3=54\).
Finding the n-th Term
- Arithmetic: \(T_n=a+(n-1)d\)
- Geometric: \(T_n=a\cdot r^{\,n-1}\)
Sum of First n Terms
Arithmetic: \(S_n=\dfrac{n}{2}\big(2a+(n-1)d\big)=\dfrac{n}{2}(a+l)\) (where \(l\) is the n-th term).
- Example: \(3+7+11+15+19\): \(n=5,\;a=3,\;l=19\Rightarrow S_5=\tfrac{5}{2}(3+19)=55\).
Geometric (\(r\neq1\)): \(S_n=a\,\dfrac{r^n-1}{r-1}\).
- Example: \(2+6+18+54\): \(a=2,\;r=3,\;n=4\Rightarrow S_4=2\cdot\dfrac{3^4-1}{3-1}=80\).
Special Sequences
- Square numbers: \(1,4,9,16,\dots\) → \(n^2\)
- Cube numbers: \(1,8,27,64,\dots\) → \(n^3\)
- Triangular numbers: \(1,3,6,10,15,\dots\) → \(T_n=\dfrac{n(n+1)}{2}\)
Identifying Patterns
- Constant difference → arithmetic.
- Constant ratio → geometric.
- Otherwise: look for squares/cubes, Fibonacci-type, or piecewise rules.
Sequences in Word Problems
- Saving a fixed amount → arithmetic.
- Repeated growth/decay by a factor → geometric.
- Design or tiling patterns → often arithmetic or triangular numbers.
Worked Examples
Example 1 (Foundation): Arithmetic n-th term
Sequence: \(5,8,11,14,\dots\). Find \(T_{10}\).
- \(a=5,\;d=3\)
- \(T_{10}=5+(10-1)\times3=32\)
Example 2 (Foundation): Geometric n-th term
Sequence: \(3,6,12,24,\dots\). Find \(T_5\).
- \(a=3,\;r=2\)
- \(T_5=3\cdot2^{4}=48\)
Example 3 (Higher): Arithmetic term & sum
Sequence: \(2,5,8,11,\dots\). Find \(T_{20}\) and \(S_{20}\).
- \(a=2,\;d=3,\;n=20\)
- \(T_{20}=2+(19)\times3=59\)
- \(S_{20}=\tfrac{20}{2}(2+59)=610\)
Example 4 (Higher): Geometric sum
Sequence: \(2,6,18,\dots\) with \(r=3\), first \(n=4\) terms.
- \(S_4=2\cdot\dfrac{3^4-1}{3-1}=80\)
Example 5 (Higher): Triangular number
Find the 7th triangular number.
- \(T_7=\dfrac{7\cdot8}{2}=28\)
Example 6 (Higher): Square number
Find the 9th square number.
- \(9^2=81\)
Example 7 (Higher): Savings (arithmetic sum)
John saves £5 more each week: \(5,10,15,\dots\). Total after 12 weeks?
- \(a=5,\;d=5,\;n=12\)
- \(S_{12}=\tfrac{12}{2}\big(2\cdot5+(11)\cdot5\big)=6\cdot65=390\) → £390
Example 8 (Higher): Geometric growth
Population triples yearly: Year 1 = 100. Find Year 5.
- \(a=100,\;r=3,\;n=5\)
- \(T_5=100\cdot3^{4}=8100\)
Example 9 (Higher): Missing term
Sequence: \(4,7,\_,13,\dots\). Find the missing term.
- Arithmetic with \(d=3\): third term \(=4+2\cdot3=10\).
Example 10 (Higher): Reverse n-th term
Arithmetic: \(2,6,10,\dots\). If \(T_n=50\), find \(n\).
- \(a=2,\;d=4\). \(50=2+(n-1)\cdot4\Rightarrow48=4(n-1)\Rightarrow n=13\).
Common Mistakes
- Mixing up arithmetic and geometric rules.
- Using the wrong n-th term or sum formula.
- Miscalculating the difference \(d\) or ratio \(r\).
- Arithmetic slips when substituting into formulas.
- Ignoring context in word problems.
Applications
- Finance: regular saving (arithmetic), compound growth (geometric).
- Population and radioactive processes (geometric growth/decay).
- Design and construction patterns (arithmetic/triangular).
- Exam modelling: predicting terms, sums, and checking rules.
Strategies & Tips
- Always identify \(a\) first, then find \(d\) or \(r\).
- Use \(T_n=a+(n-1)d\) or \(T_n=a r^{\,n-1}\) carefully.
- For sums, choose the correct formula (arithmetic vs geometric).
- Verify with a few actual terms to catch mistakes early.
- Practise word problems to connect sequences with context.
Summary / Call-to-Action
Sequences turn patterns into precise formulas. By mastering arithmetic and geometric rules, n-th terms, and sums, you can predict values, fill gaps, and model real-world growth and savings confidently.
- Drill identifying \(a\), \(d\), and \(r\).
- Practise n-th terms, missing terms, and sums.
- Apply to finance and growth contexts for deeper understanding.