Page 53 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

2. Investigating Arithmetic and Geometric Sequences and Series
By the end of this course, students will:
2.1 identify sequences as arithmetic, geometric, or neither, given a numeric or algebraic representation
2.2 determine the formula for the general
term of an arithmetic sequence [i.e.,
tn = a + (n – 1)d ] or geometric sequence (i.e., tn = ar n – 1), through investigation using a variety of tools (e.g., linking cubes, algebra tiles, diagrams, calculators) and strategies (e.g., patterning; connecting the steps in a numerical example to the steps in the algebraic development), and apply the formula to calculate any term in a sequence
2.3 determine the formula for the sum of an arithmetic or geometric series, through inves- tigation using a variety of tools (e.g., linking cubes, algebra tiles, diagrams, calculators) and strategies (e.g., patterning; connecting the steps in a numerical example to the steps in the algebraic development), and apply
the formula to calculate the sum of a given number of consecutive terms
Sample problem: Given the following array built with grey and white connecting cubes, investigate how different ways of determin- ing the total number of grey cubes can be used to evaluate the sum of the arithmetic series 1 + 2 + 3 + 4 + 5. Extend the series, use patterning to make generalizations for finding the sum, and test the generalizations for other arithmetic series.
By the end of this course, students will:
3.1 make and describe connections between simple interest, arithmetic sequences, and linear growth, through investigation with technology (e.g., use a spreadsheet or graphing calculator to make simple interest calculations, determine first differences in the amounts over time, and graph amount versus time)
Sample problem: Describe an investment that could be represented by the function f(x) = 500(1 + 0.05x).
3.2 make and describe connections between compound interest, geometric sequences, and exponential growth, through investiga- tion with technology (e.g., use a spreadsheet to make compound interest calculations, determine finite differences in the amounts over time, and graph amount versus time)
Sample problem: Describe an investment that could be represented by the function
f(x) = 500(1.05)x.
3.3 solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, FV ),
the principal, P (also referred to as
present value, PV ), or the interest rate per compounding period, i, using the compound interest formula in the form A=P(1+i)n [orFV=PV(1+i)n]
Sample problem: Two investments are available, one at 6% compounded annually and the other at 6% compounded monthly. Investigate graphically the growth of each investment, and determine the interest earned from depositing $1000 in each investment for 10 years.
3.4 determine, through investigation using technology (e.g., scientific calculator, the TVM Solver on a graphing calculator, online tools), the number of compounding periods, n, using the compound interest formula in the formA=P(1+i)n [orFV=PV(1+i)n]; describe strategies (e.g., guessing and check- ing; using the power of a power rule for exponents; using graphs) for calculating this number; and solve related problems
3. Solving Problems Involving Financial Applications
2.4 solve problems involving arithmetic and geo- metric sequences and series, including those arising from real-world applications
DISCRETE FUNCTIONS
51
Functions
MCR3U