Algebra II End-of-Course Exam Content Standards—Module: Sequences and Series
This module addresses the patterns in arithmetic and geometric sequences and series. Students are expected to apply the formulas for finding the nth term of a sequence or series, the nth partial sum of finite series, and the infinite sum of a geometric series when it exists. General iterative relationships and recursive models are applied to patterns and problems. There will be a variety of types of test items including some that cut across the objectives in this standard and require students to make connections and solve rich contextual problems, where appropriate.
I1. Arithmetic and geometric sequences and series
a. Represent the general term of an arithmetic or geometric sequence and use it to generate the sequence or determine the value of any particular term.
- Use and interpret sigma notation, ∑, to indicate summation.
- Recognize that the sequence defined by: First term = 5; each term after the first is 6 more than the preceding term as the sequence whose first seven terms are 5, 11, 17, 23, 29, 35, and 41.
b. Represent partial sums of an arithmetic or geometric sequence and determine the value of a particular partial sum or sum of a finite sequence.
- Use the formulas for sums of finite arithmetic and geometric series to find partial sums or the sum of n terms.
c. Recognize when an infinite geometric sum can be determined and determine the sum when possible.
- Determine that the sum exists when the common ratio, r, is between 1 and -1. (|r| < 1)
- To find the infinite sum, when it exists, use the formula
, where a represents the first term or the geometric series and r is the common ratio.
d. Convert the recursive model for linear growth (a1 = a, an+1 = an + d, where a is the first term and d is the constant difference) to a closed linear form (an = a + (n - 1)d).
e. Convert the recursive model of geometric growth (p1 = a, pn+1 = rpn, where a is the first term and r is the constant growth rate) to a closed exponential form (pn = arn-1).
f. Recognize, express, and solve problems that can be modeled using a finite geometric series. Interpret their solutions in terms of the context of the problem.
- Possible applications are home mortgage problems and other compound interest problems.
- This includes using and interpreting appropriate units of measurement and precision for the given application.
I2. Other types of iteration and recursion
a. Use recursion to generate and describe, analyze, and interpret patterned relationships other than arithmetic or geometric sequences.
- Analyze and explain the iterative steps in standard algorithms for arithmetic.
- Use recursion to generate and describe patterned relationships.
- Analyze sequences produced by recursive calculations using spreadsheets.
Example: The result of repeatedly squaring a number between -1 and 1 appears to approach zero; the result of repeatedly squaring a number less than -1 or greater than 1 appears to continue to increase; determine empirically how many steps are needed to produce 4-digit accuracy in square roots by iterating the operations divide and average.
- Describe the factorial function or the Fibonacci sequence recursively.
b. Use iterative methods to solve problems.
- Display the effect of iteration using a spreadsheet.
Example: Use a spreadsheet to display and compare the effect of compound interest on loans or investments at different rates of interest; identify the diminishing effect of increasing the number of times per year that interest is compounded.