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13.1 Arithmetic and Geometric Sequences Explicit Formulas 13.1 Arithmetic and Geometric Sequences Objectives: Identify Arithmetic and Geometric sequences 2. Complete missing information associated with sequences 3. Define formulas for sequences 1. Vocabulary: Sequence, term, Arithmetic, Geometric Learning The Lingo A "sequence" an ordered list of numbers; the numbers in this ordered list are called "elements" or "terms". Learning the Lingo A sequence may be named or referred to as "A" or "An". The terms of a sequence are usually named something like "ai" or "an", with the subscripted letter "i" or "n" being the "index" or counter. An = a1 a2 a3 a4 a5 Arithmetic Sequences The two simplest sequences to work with are arithmetic and geometric sequences. An arithmetic sequence goes from one term to the next by always adding (or subtracting) the same value. For instance: Add 3 to each term Example One: 2, 5, 8, 11, 14,... Example Two: 7, 3, –1, –5,... Subtract 4 to each term Learning The Lingo The number added (or subtracted) at each stage of an arithmetic sequence is called the "common difference" d, because if you subtract (find the difference of) successive terms, you'll always get this common value. Writing Arithmetic Sequences an = a1 + (n – 1)d Example – Your Turn Find the common difference and the next term of the following sequence: 3, 11, 19, 27, 35,... The difference is always 8, so d = 8. Then the next term is 35 + 8 = 43. Examples: Find a formula for an and find the 10th term 2,6,10,14,18,… 17,10,3,-4,-11,-18,… Examples: Find the n-th term (formula) of the arithmetic sequence having a4 = 93 and a8 = 65. Since a4 and a8 are four places apart, then I know from the definition of an arithmetic sequence that a8 = a4 + 4d. 65 = 93 + 4d –28 = 4d –7 = d 93 = a + 3(–7) 93 + 21 = a 114 = a OR 65 = a + 7(–7) 65 + 49 = a 114 = a Geometric Sequences A geometric sequence goes from one term to the next by always multiplying (or dividing) by the same value. For instance: Example One: 1, 2, 4, 8, 16,... Multiply by 2 at each step Example Two: 81, 27, 9, 3, 1, 1/3,... Divide by 3 at each step Learning The Lingo The number multiplied (or divided) at each stage of a geometric sequence is called the "common ratio" r, because if you divide (find the ratio of) successive terms, you'll always get this common value. Writing Geometric Sequences an = a1 (n – 1) r Example – Your Turn! Find the common ratio and the seventh term of the following sequence: 2/9, 2/3, 2, 6, 18,... The ratio is always 3, so r = 3. Then the sixth term is (18)(3) = 54 and the seventh term is (54)(3) =162 Examples: Find a formula for an and find the 10th term 1,3,9,27,81,… 64,-32,16,-8,4,… Example Find the n-th (formula) of the geometric sequence with a5 = 5/4 and a12 = 160. These two terms are 12 – 5 = 7 places apart, so, from the definition of a geometric sequence, I know that 7 160 = (5/4)(r ) 128 = r7 2=r 4 5/4 = a(2 ) = 16a 5/64 = a a12=a5r7 11 160 = a(2 ) = 2048a OR 160/2048 =5/64= a Homework: Textbook: p. 477 #17-21, 29, 33 Practice WB p. 91 # 4 No, multiply is Geometric Practice WB p. 98 # 3 an=3(1/4)(n-1) or an=12(1/4)(n) a8=3/47 or a8=0.0001831 Practice WB p. 91 # 8 an= 1 + (n – 1)-5 or an= - 5n + 6 a30= -144 Practice WB p. 97 # 3 an=1024(+/-1/4)(n-1) an=4096(+/-1/4)(n) Practice WB p. 91 # 12 an= -81 + (n – 1) 11 or 11n + 92