Sectio.2 Arithmetic Sequeces ad Series -.2 Arithmetic Sequeces ad Series Arithmetic Sequeces Arithmetic Series Key Terms: arithmetic sequece (arithmetic progressio), commo differece, arithmetic series A (or ) is a sequece i which each term after the first differs from the precedig term by a fixed costat, called the. If the commo differece of a arithmetic sequece is d, the by the defiitio of arithmetic sequece,, for every positive iteger i the domai of the sequece. EXAMPLE Fidig the Commo Differece Fid the commo differece, d, for the arithmetic sequece 9, 7, 5, 3,, EXAMPLE 2 Fidig Terms Give a ad d Fid the first five terms for each arithmetic sequece. (a) The first term is 7, ad the commo (b) a 2, d 5 differece is 3. th Term of a Arithmetic Sequece I a arithmetic sequece with first term a ad commo differece d, the th term, a, is give by the followig. a a d EXAMPLE 3 Fidig Terms of a Arithmetic Sequece Fid a 3 ad a for the arithmetic sequece 3,, 5, 9,.
EXAMPLE 4 Fidig Terms of a Arithmetic Sequece Fid a ad a 8 for the arithmetic sequece havig a2 9 ad a3 5. EXAMPLE 5 Fidig the First Term of a Arithmetic Sequece Suppose that a arithmetic sequece has a8 6 ad a6 40. Fid a. EXAMPLE 6 Fidig the th Term from a Graph Fid a formula for the th term of the sequece a show i the figure. What are the domai ad rage of this sequece? Arithmetic Series: The sum of the terms of a arithmetic sequece is a. Sum of the First Terms of a Fiite Arithmetic Sequece If a arithmetic sequece has first term a ad commo differece d, the the sum S of the first terms is give by these formulas. S a a, or S 2a d 2 2 The first formula is used whe the first ad last terms are kow; otherwise, the secod formula is used. EXAMPLE 7 Usig the Sum Formulas (a) Evaluate S 2 for the arithmetic sequece 9, 5,, 3, 7,.
Sectio.2 Arithmetic Sequeces ad Series -3 (b) Use a formula for S to evaluate the sum of the first 60 positive itegers. ***EXAMPLE 8 Usig the Sum Formulas The sum of the first 7 terms of a arithmetic sequece is 87. If a7 3, fid a ad d. EXAMPLE 9 Usig Summatio Notatio Evaluate each sum. (a) 0 (4i 8) ***(b) i 9 k 3 (4 3 k)
.3 Geometric Sequeces ad Series Geometric Sequeces Geometric Series Ifiite Geometric Series Auities Key Terms: geometric sequece (geometric progressio), commo ratio, geometric series, auity, future value of a auity A (or ) is a sequece i which each term after the first is obtaied by multiplyig the precedig term by a fixed ozero real umber, called the. We fid the commo ratio by choosig ay term except the first ad dividig it by the precedig term. th Term of a Geometric Sequece I a geometric sequece with first term a ad commo ratio r, the th term, a, is give by the followig. a ar EXAMPLE Fidig the th Term of a Geometric Sequece Imagie workig at a job where you were paid i a uorthodox maer: $0.0 the first day, $0.02 the secod day, $0.04 the third day, $0.08 the fourth day, ad so o, with your wages doublig each day. How much will be eared o day 20? EXAMPLE 2 Fidig Terms of a Geometric Sequece Fid a 5 ad a for the geometric sequece 4, 2, 36, 08,. Reflect: What is the major differece betwee arithmetic sequeces ad geometric sequeces? ***EXAMPLE 3 Fidig Terms of a Geometric Sequece Fid r ad a for the geometric sequece with third term 20 ad sixth term 60.
EXAMPLE 4 Modelig a Populatio of Fruit Flies A populatio of fruit flies is growig i such a way that each geeratio is.5 times as large as the last geeratio. Suppose there are 00 isects i the first geeratio. How may would there be i the fourth geeratio? Geometric Series: A is the sum of the terms of a geometric sequece. Sum of the First Terms of a Fiite Geometric Sequece If a geometric sequece has first term a ad commo ratio r, the the sum of the first terms is give by the followig. S a r, r where r EXAMPLE 5 Fidig the Sum of the First Terms At the begiig of this sectio we asked you to imagie workig at a job where you were paid $0.0 the first day, $0.02 the secod day, $0.04 the third day, $0.08 the fourth day, ad so o, with your wages doublig each day. How much will you have eared altogether after 20 days? EXAMPLE 6 Fidig the Sum of the First Terms Fid 6 i 23. i Sum of the Terms of a Ifiite Geometric Sequece The sum of the terms of a ifiite geometric sequece with first term a ad commo ratio r, where r, is give by the followig. a S r If r, the terms of the sequece will ot have a sum.
EXAMPLE 7 Evaluatig a Ifiite Geometric Series Evaluate. 3 9 27 EXAMPLE 8 Evaluatig Ifiite Geometric Series Fid each sum. (a) i i 3 (b) 4 2 i 3 5 i