LCHL Arithmetic & Geometric Sequences & Series
• N: the set of natural numbers, N = {1,2,3,4…} • Z: the set of integers, including 0 • Q: the set of rational numbers
I should be able to:
Ordinary Level
– recognise irrational numbers and appreciate that R ≠ Q
– work with irrational numbers
– revisit the operations of addition, multiplication, subtraction and division in the following domains: ka3.1.1a ka3.1.1b ka3.1.1c
• N of natural numbers • Z of integers • Q of rational numbers • R of real numbers and represent these numbers on a number line
– investigate the operations of addition, multiplication, subtraction and division with complex numbers c in rectangular form a+ib
– illustrate complex numbers on an Argand diagram
– interpret the modulus as distance from the origin on an Argand diagram and calculate the complex conjugate
– develop decimals as special equivalent fractions strengthening the connection between these numbers and fraction and place-value understanding
– consolidate their understanding of factors, multiples, prime numbers in N
– express numbers in terms of their prime factors
– appreciate the order of operations, including brackets
– express non-zero positive rational numbers in the form a x10n, where n ∈ Z and 1 ≤ a < 10 and perform arithmetic operations on numbers in this form
– appreciate that processes can generate sequences of numbers or objects
– investigate patterns among these sequences
– use patterns to continue the sequence
– generalise and explain patterns and relationships in algebraic form
– recognise whether a sequence is arithmetic, geometric or neither
– find the sum to n terms of an arithmetic series
Higher Level
– geometrically construct √2 and √3
– prove that √2 is not rational
– calculate conjugates of sums and products of complex numbers
– verify and justify formulae from number patterns
– investigate geometric sequences and series
– prove by induction
• simple identities such as the sum of the first n natural numbers and the sum of a finite geometric series
• simple inequalities such as n! > 2n Proj Maths Site, 2n > n2 (n ≥ 4) Proj Maths Site, (1+ x)n ≥ 1+nx (x > –1) Proj Maths Site
• factorisation results such as 3 is a factor of 4n–1 Proj Maths Site 1 Proj Maths SIte 2
– apply the rules for sums, products, quotients of limits
– find by inspection the limits of sequences such as :
lim n
n →∞ n+1
lim rn |r|<1
n →∞
– solve problems involving finite and infinite geometric series including applications such as recurring decimals and financial applications, e.g. deriving the formula for a mortgage repayment
– derive the formula for the sum to infinity of geometric series by considering the limit of a sequence of partial sums