Page 52 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

GGrraade11,, University Preparraattioionn
1. demonstrate an understanding of recursive sequences, represent recursive sequences in a variety of ways, and make connections to Pascal’s triangle;
2. demonstrate an understanding of the relationships involved in arithmetic and geometric sequences and series, and solve related problems;
3. make connections between sequences, series, and financial applications, and solve problems involving compound interest and ordinary annuities.
1. Representing Sequences
C. DISCRETE FUNCTIONS OVERALL EXPECTATIONS
By the end of this course, students will:
SPECIFIC EXPECTATIONS
By the end of this course, students will:
1.1 make connections between sequences and discrete functions, represent sequences using function notation, and distinguish between a discrete function and a continuous function [e.g., f(x) = 2x, where the domain is the set of natural numbers, is a discrete linear function and its graph is a set of equally spaced points; f(x) = 2x, where the domain is the set of real numbers, is a continuous linear function and its graph is a straight line]
1.2 determine and describe (e.g., in words; using flow charts) a recursive procedure for gen- erating a sequence, given the initial terms (e.g., 1, 3, 6, 10, 15, 21, ...), and represent sequences as discrete functions in a variety of ways (e.g., tables of values, graphs)
1.3 connect the formula for the nth term of a sequence to the representation in function notation, and write terms of a sequence given one of these representations or a recursion formula
1.4 represent a sequence algebraically using a recursion formula, function notation, or the formula for the nth term [e.g., represent 2, 4, 8, 16, 32, 64, ... as t1 = 2; tn = 2tn – 1, as
f(n)=2n,orastn =2n,orrepresent 21, 32, 43,
4, 5, 6,...ast1= 1;tn=tn–1 + 1 567 2 n(n+1)
asf(n)= n ,orastn= n ,wheren n+1 n+1
,
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
is a natural number], and describe the infor- mation that can be obtained by inspecting each representation (e.g., function notation or the formula for the nth term may show the type of function; a recursion formula shows the relationship between terms)
Sample problem: Represent the sequence
0, 3, 8, 15, 24, 35, ... using a recursion formula, function notation, and the formula for the nth term. Explain why this sequence can be described as a discrete quadratic function. Explore how to identify a sequence as a discrete quadratic function by inspecting the recursion formula.
1.5
determine, through investigation, recursive patterns in the Fibonacci sequence, in related sequences, and in Pascal’s triangle, and
represent the patterns in a variety of ways (e.g., tables of values, algebraic notation)
1.6
determine, through investigation, and describe the relationship between Pascal’s triangle and the expansion of binomials,
and apply the relationship to expand bino- mials raised to whole-number exponents [e.g., (1 + x)4, (2x –1)5, (2x – y)6, (x2 + 1)5]
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