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Introduction to sequences A sequence is an ordered list of numbers: 3, 6, 9, 12, .... 1, 3, 5, 7, ... A sequence can be thought of as a list of numbers written in a definite order: a1, a2, a3, ..., an, ... The number a1 is called the first term, a2 is the second term, and in general an is the nth term. The terms of a sequence often follow a particular pattern. In those instances, we can determine the general term that expresses every term of the sequence. For example: 1.Sequence General Term 3, 6, 9, 12,... 1,3,5,7,... 2,4,8,16,... 3n 2n-1 2n 2.- Given an=3n+2, find a1, a2, a3 and a4. By substituting n=1,2,3,4 in the general term of the sequence, we obtain: a1=3(1)+2=3+2=5 a2=3(2)+2=6+2=8 a3=3(3)+2=9+2=11 a4=3(4)+2=12+2=14 3.- Find the general term for the following sequences: a) 4, 8, 12, 16,... b) , , , , ... c) , , , , ... Solution: a) 4, 8, 12, 16,... is the same as: b) , , , c) , , , , ... , ... is the same as: 1·4, 2·4, 3·4, 4·4,... The general term is: an=4n , , , The general term is: cn= , ...The general term is: bn= Arithmetic sequences An arithmetic sequence is a sequence of numbers each of which, after the first, is obtained by adding to the preceding number a constant number called the common difference. Thus, 3, 8, 13, 8, 23,... is an arithmetic sequence because each term is obtained by adding 5 to the preceding number. If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then: a2=a1+d a3=a2+d=a1+2d a4=a3+d=a1+3d an=a1+(n-1)d To solve exercises using arithmetic sequences you need the following formulas: • The nth term: an= a1+ (n-1) d where: a1 = the first term of the sequence d = common difference (Remember you can obtain it using d=an-an-1 or n = number of terms an = nth term Example: Given the arithmetic sequence 11, 7, 3,... Find the 301st term, a301 an= a1+ (n-1) d a301= 11+ (300-1)·(-4)=-1189 Sum of an arithmetic sequence An arithmetic series is the sum of an arithmetic sequence. That is, a1+a2+a3+...+an=a1+a1+d+a1+2d+a1+3d+...+a1+(n-1)d The series 3 + 7 + 11 + 15 + 19 + ... + 99 is an arithmetic series which first term is a1=3 and the common difference is d=4. To solve exercises using arithmetic sequences you need the following formulas: • The nth term: an= a1+ (n-1) d .) • The sum of the first n terms: where: a1 = the first term of the sequence d = common difference (Remember you can obtain it using d=an-an-1 or n = number of terms an = nth term Sn = sum of the first n terms .) Sometimes we use: Example: Find the sum of the first 40 terms, s40 of 3, 7, 11, 15,... Geometric sequences A geometric sequence is a sequence of numbers each of which, after the first, is obtained by multiplying the preceding number a constant number called the common rate. a1 a2=a1·r a3=a2·r=a1·r2 a4=a3·r=a1·r3 ... an=a1·rn-1 3, 6, 12, 24, 48,... is a geometric sequence because each term is obtained by multiplying the preceding number by 2. To solve exercises using geometric sequences you need the following formula: The nth term: Where: a1 = the first term of the sequence r = common rate n = number of terms an = nth term an=a1·rn-1 Example: Given 27, -9, 3, -1, ... Find an and a8 Using an=a1·rn-1 Example: Given a geometric sequence with a2=-10 and a5=-80. Find an. an=a1·rn-1, so we need to find a1. To find it we use the next system of equations: solving by substitution: , and a1=-5. That is an=-5·2n-1 Geometric series A geometric series is the sum of a geometric sequence. That is, a + ar + ar2 + ar3 + ar4 + ... where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start value. 3 + 6 + 12 + 24 + 48 +... is a geometric series because each term is obtained by multiplying the preceding number by 2. To solve exercises using geometric sequences you need the following formulas: n-1 • The nth term: an=a1·r • The sum of the first n terms: • The sum of an infinite geometric serie: only if |r|<1. where: a1 = the first term of the sequence r = common rate n = number of terms an = nth term Sn = sum of the first n terms S= sum of an infinite geometric series Example: Given 1/6, 1/2, 3/2, 9/2, find S7 This is a geometric series with a common rate r=3. Recursion Sequences A recursive sequence is an ordered list of numbers defined by a starting value and a rule. You generate the sequence by applying the rule to the starting value, applying the rule again to the resulting value, and then repeating this process. A recursive formula always has two parts: 1.- the starting values. 2.- the recursion equation for an as a function of the terms before it. Example: 1, 1, 2, 3, 5, 8, ... is a recursive sequence, where: a1=1 a2=1 an=an-1+an-2 (It is well known as The Fibonacci Sequence) Example: Given: a1=1, a2=2,y an=2an-1+an-2 Find the next three terms. a3=2a2+a1=2(2)+1=5 a4=2a3+a2=2(5)+2=12 a5=2a4+a3=2(12)+5=29