Algebraic Expressions, Sequences and Series
Algebra represents patterns and relationships using symbols. Sequences and series then organise patterns of numbers and their sums.
Learning objectives
After studying these notes, learners should be able to define the main terms, apply the central rules and formulas, solve structured examples and explain each answer clearly.
Key ideas
- Expansion and factorisation
- Algebraic fractions
- Linear and quadratic equations
- Completing the square
- Arithmetic progressions
- Geometric progressions
Essential rules and formulas
- For ax^2+bx+c=0, x = (-b plus or minus sqrt(b^2-4ac))/(2a)
- Arithmetic term: u_n = a+(n-1)d
- Arithmetic sum: S_n = n/2[2a+(n-1)d]
- Geometric term: u_n = ar^(n-1)
Worked example
For 5, 8, 11, … the first term is 5 and common difference is 3, so u_n = 5 + 3(n-1) = 3n+2.
Study method
Begin every problem by listing the information supplied and identifying the rule that connects it to the unknown quantity. Keep algebraic steps on separate lines, include units where required and check that the final result is reasonable.
Practice questions
- Factorise 6x^2-x-2.
- Solve 2x^2-5x-3=0.
- Find the sum of the first 20 terms of 4, 7, 10, …
Revision checklist
- Can you state each definition without looking at the notes?
- Can you select the correct formula and substitute values accurately?
- Can you explain the reasoning behind each step?
- Can you solve a similar examination question independently?
