Chapter 12: Sequences and Series
A sequence is an ordered list of numbers following a pattern. A series is the sum of terms of a sequence. We study arithmetic sequences (constant difference), geometric sequences (constant ratio), and their sums. The binomial theorem expands (a+b)โฟ exactly using Pascal's triangle and combinatorial coefficients.
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Sections
12.1Sequences and Summation Notation
A sequence is a function whose domain is the natural numbers. Terms can be defined by explicit formulas or recursively. Sigma notation ฮฃ provides a compact way to write sums, with algebraic properties that simplify computation. Standard formulas give closed-form expressions for sums of integers and squares.
12.2Arithmetic Sequences
An arithmetic sequence has a constant difference d between consecutive terms, producing linear growth. The nth term formula and sum formula are two essential tools. Given any two terms, the common difference and first term can be recovered. Applications include salary schedules, seating arrangements, and installment plans.
12.3Geometric Sequences
A geometric sequence has a constant ratio r between consecutive terms, producing exponential growth or decay. Finite and infinite sum formulas apply, with the infinite series converging only when |r| < 1. Applications include compound interest, population models, and repeating decimals.
12.4The Binomial Theorem
The Binomial Theorem expands (a+b)โฟ without multiplying out n factors. Coefficients come from Pascal's triangle or the combination formula C(n,k) = n!/(k!(nโk)!). The (k+1)-th term is C(n,k)aโฟโปแตbแต, allowing any specific term to be found directly.
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