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Section 11.1 Sequences and Summation Notation A sequence is a set of numbers written in a specific order: a1, a2, a3,.......,an,..... The number a1 is called the first term, a2 is the second, etc. Definition of a sequence A sequence is a function f whose domain is the set of natural numbers. The values f(1), f(2), f(3), ... are called the terms of the sequence Finding the terms of a sequence Find the first five terms and the 50th term of the sequence. an = n2 3 an = (1)n(3n 1) Recursive Sequences A recursive sequence is one where the nth term may depend on some or all of the preceding terms. Example: Find the first five term of the recursive sequence defined by an = 3an1 + 2, = 5 Partial Sums of a Sequence For the sequence a1, a2, a3,...an,... the partial sums are: S1 = a1 S2 = a1+ a2 S3 = a1 + a2 + a3 . . . Sn = a1+ a2 + a3 + a4 + ....+ an S1 is called the first partial sum, S2 is second partial sum and so on. The sequence S1, S2, S3,...., Sn is called the sequence of partial sums. Example: Find the first four partial sums of the sequence given by an = 3n 4 Sigma Notation the sum of the first n terms can be written using summation notation or sigma notation. n ak = a1 + a2 + a3 +.....+ an k = 1 The left side is read "the sum of ak from k = 1 to k = n." k is called the index of summation and you need to replace k in the expression after the sigma by the integers 1, 2, 3, 4, ..., n and add the resulting expressions Example: Find each sum: 5 6 3k 4k2 k = 1 k = 3 Using the calculator to find sums You will need to type into your calculator the following expression: sum(seq(expression you are finding sum of, variable being used, start value, end value, increment)) found under 2nd list, MATH found under 2nd list, OPS 100 Example: Find the following sum: k = 4 Properties of Sums n n n (ak + bk) = ak + bk 1. k = 1 k = 1 k = 1 n n n 2. (ak bk) = ak bk k = 1 k = 1 k = 1 n n 3. cak = c ak k = 1 k = 1 k3 23