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MCR3U1 U7L1 DISCRETE FUNCTIONS ~ SEQUENCES AND SERIES (Arithmetic Sequences) PART A ~ INTRODUCTION A numerical sequence is a set of numbers arranged in an order. A sequence may have a finite or an infinite number of terms. Ex. 1, 4, 9, 16, 25, 36, 49, … The notation used to represent the terms of a sequence is: ๐ก1 → the 1๐ ๐ก term ๐ก2 → the 2๐๐ term ๐ก๐ → the ๐๐กโ term Sometimes a pattern can lead to a general rule for determining the terms of a sequence. This rule is called the formula for the ๐๐๐ term or the general term (๐๐ ). Ex. ๐ก๐ = ๐2 PART B ~ ARITHMETIC SEQUENCES arithmetic sequence: a sequence where the difference between consecutive terms is a constant common difference (๐ ): the amount that is added (or subtracted) to each term of a sequence to get the next term of the sequence Ex. 1, 5, 9, 13, … or 1, –3, –7, –11, … DERIVING THE GENERAL TERM (๐๐ ) ๐ก1 , ๐ก2 , ๐ก3 , ๐ก4 , ๐ก5 , … ๐ก๐ GENERAL TERM of an ARITHMETIC SEQUENCE ๐ก๐ = ๐ + (๐ − 1)๐ where ๐ is ๐ก1 (the first term) ๐ is the number of terms ๐ is the common difference ( ๐ = ๐ก๐ − ๐ก๐−1 ) MCR3U1 U7L1 PART C ~ EXAMPLES Ex ๏ Determine which of the following are arithmetic sequences. If they are, state the common difference (๐). a) 2, 7, 12, 17, … b) 6, 10, 12, 13, … c) 5, 2, –1, –4, … Ex ๏ Given ๐ก1 = 7 and ๐ = 3 for an arithmetic sequence, write the first four terms of the sequence and state the general term (๐ก๐ ). Ex ๏ State the general term and then calculate ๐ก15 for the given arithmetic sequence: 3, 7, 11, 15, … Ex ๏ Calculate the number of terms in the given arithmetic sequence: 9, 4, –1, …, –146 MCR3U1 U7L1 Ex ๏ The third term of an arithmetic sequence is –11 and the seventh term is –27. Determine the first three terms of the sequence. Ex ๏ Boxes are stacked in a store display in the shape of a triangle. The number of boxes in the rows form an arithmetic sequence. There are 41 boxes in the third row from the bottom and 23 boxes in the twelfth row from the bottom. a) b) Ex ๏ Determine the values of ๐ and ๐ for this sequence. Determine the maximum possible number of rows of boxes. Determine the value of ๐ฅ that makes the given sequence arithmetic: 2๐ฅ + 9, −4๐ฅ, −3๐ฅ + 5, … HOMEWORK: p.424–425 #1, 3, 4, 8(parts i & iii), 10, 13ace, 15