Sequences and Series

Definition: A sequence is an ordered list of numbers separated by commas. There are two basic types: arithmetic and geometric. A term is each number in the sequence. An example of a sequence is {2, 4, 6, 8, ..., n}. The position of each number is important, ie. 1st term, 2nd term, 3rd term or nth term. An infinite sequence is a sequence that continues indefinitely so n is infinity. An Arithmetic Sequence is generated when a constant number called the common difference(d) is added to each term to produce the next term. "d" may be positive or negative so a sequence can increase or decrease. {1,3,5,7,...}, {5,8,11,14,...}, and {20,17,14,11,8} are sample arithmetic sequences. {1,3,9,27,...} is NOT arithmetic. A very important formula exists for working with an Arithmetic Sequence... an = a1 + (n - 1)d where anrepresents the nth term, a1 represents the first term, n represents the total number of terms, and d represents the common difference. For {1, 3, 5, 7, ...}, a1 = 1, and d = 2. Substituting this info into the formula we generate a "rule" which describes this particular sequence. an = 1 + (n-1)2 = 2 + 2n - 1 = 2n -1 This resembles the Linear Functions of form y = mx + b that we studied earlier. When an arithmetic sequence is graphed on a coordinate plane, a series of distinct points which are aligned result. The sequence above would contain the points (1,1), (2,3), (3,5), (4,7),....(n, 2n-1). There is no line connecting the points. We also have a rule which will allow us to find the number present in any term, like a53. a53 = 2(53) - 1 = 106 -1 = 105 The 53rd term in the arithmetic sequence {1, 3, 5, 7, ...} is 105. You might like to try one yourself...Find the rule for an for the sequence {-2, 4, 10, 16, ...}, then find a31 Occasionally you may be asked to use the formula above to answer another question like "Find the first term of the arithmetic sequence with common difference = 2 and a31 = 52". The procedure is to substitute the known values into the formula and solve algebraically for the requested value. An Arithmetic Series (denoted Sn) resembles a sequence except that the commas are replaced by "+". Arithmetic Series are really Arithmetic Sums. Note: Sums of terms in a sequence are considered to be "partial sums" since they are the sum of a specified number of terms. It is not possible to find a sum for an infinite number of terms. There is a formula for finding partial sums for arithmetic sequences... Sn = (n/2)(a1+an) Consider {5,8,11,14,17,...}. Find S11, ie. find the sum of the first 12 terms. Substituting into the formula we get S12 = (12/2)(5 + an) We need to take a little side trip and find an using our first formula when n = 12. After substituting we get an = 38. So....S12 = (12/2)(5+38) = 6(43)=258. 12 is not such a terribly large number...you could actually list the first 12 terms out and then find the sum. However, if the sum requested were 112, the formula would be a real time saver!      Sigma Notation is a symbolic way to request a partial sum. Here is a sample problem and solution: The problem is asking for the sum of the terms from #1 through #19 generated by the rule 3n + 1. We must find all the values required for the execution of the formula. So... a1 = 3(1) = 1 = 3 + 1 = 4 and an becomes a19 = 3(19) + 1 = 57 + 1 = 58. Now we have all the necessary values to substitute and... S19 = (19/2)(4 + 58) = (9.5)(62) = 589 This concept must be thoroughly understood before progressing to the lesson on Geometric Sequences and Series. The basic terminology is the same but the formulas are different and the graphs look different.

[TOC] [Mathematical "Models"] [Functions] [Probability] [Direct Variation] [Solving Linear Equations] [Analyzing Equations of Lines] [One Variable Inequalities] [Arithmetic Sequences] [Geometric Sequences] [Irrational Numbers] [Complex Numbers] [Quadratic Functions] [Solving Quadratic Equations] [Conic Sections] [Variation] [Exponents and Roots] [Solving Radical Equations] [Function Operations] [Polynomial Functions] [Rational Expressions] [Rational Functions] [Solving Rational Equations] [Exponential Functions] [Logarithmic Functions]