ARITHMETIC SEQUENCES

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Samuel Cobb
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1 ARITHMETIC SEQUENCES

2 additio subtractio multiplicatio (product) value eve umbers odd umbers sequece term of the sequece arithmetic progressio(sequece) recurrece formula commo differece d cosecutive series (sum of the terms) properties average a a a 2 A=C+4 сложение вычитание умножение значение четные числа нечетные числа последовательность член последовательности арифметическая прогрессия рекуррентная формула разница последовательный сумма членов свойства среднее арифметическое egative a geeral term a-sub-two, the secod term a is equal to c plus four

3 After completig this tutorial, you should be able to: -Kow what a arithmetic sequece is. -Fid the th term of a arithmetic sequece. -Fid the arithmetic series -Solve the problems of a arithmetic sequece. -Lear ew math termiology

4 A sequece is a ordered list of umbers. sequece of positive eve umbers 2, 4, 6, 8, a 2 sequece of positive odd umbers 1, 3, 5, 7, a 21

5 A arithmetic sequece goes from oe term to the ext by always addig (or subtractig) the same value. a a d 1 (recurrece formula) The umber added (or subtracted) at each stage of a arithmetic sequece ad sice this umber is commo to all cosecutive pairs of terms, it is called the "commo differece" d d a a 1 The commo differece ca be foud by subtractig two cosecutive terms of the sequece.

6 Fid the commo differece for this arithmetic sequece 5, 9, 13, Fid the commo differece for the arithmetic sequece whose formula is a = This sequece a arithmetic progressio 20, 17, 14, 11, 8, 5, 2, -1, -4, fid the first term ad the commo differece.

7 If commo differece d<0 it s decreasig progressio If commo differece d>0 it s icreasig progressio

8 N-TH OR GENERAL TERM OF AN ARITHMETIC SEQUENCE Sice a a d where is 1 a a d 2 1 a a d a d d a 2d the first term of the sequece ad d is the commo differece. a a d a 1

9 EX: WRITE A FORMULA FOR THE N-TH TERM OF THE ARITHMETIC SEQUENCE -10, -5, 0, 5,... We will use the -th term formula for a arithmetic sequece,, to help a a d us with this problem. Basically we eed to fid the first term of the sequece ad the commo differece, d. a1 10 d a2 a1 5 ( 10) a 10 5 ( 1)

10 Some useful formulae ad properties: a a a a 1 1 a ex: a a a a a d( k) k 2 a a 4 6 a a 5d

11 Fid the commo differece ad the ext term of the followig sequece: 3, 11, 19, 27, 35,... Fid the 9-th term of the sequece: 3, 7, Fid the -th term ad the first three terms of the arithmetic sequece havig a 4 = 93 ad a 8 = 65. Fid the -th term ad the first three terms of the arithmetic sequece havig a 6 = 5 ad d = 3/2. Fid a 7 for a arithmetic sequece where are a 1 = 3x ad d = -x. Fid the umber of terms i the sequece 7, 10, 13,..., 55.

12 additio subtractio multiplicatio (product) value eve umbers odd umbers sequece term of the sequece arithmetic progressio(sequece) recurrece formula commo differece d cosecutive series (sum of the terms) properties average a a a 2 A=B+3

13 ARITHMETIC SERIES A series are created by addig terms i the sequece. It is a sum of the terms of a sequece. The sum of a arithmetic series is foud by multiplyig the umber of terms times the average of the first ad last terms. S a 1 2 a

14 EX: FIND THE SUM OF THE FIRST 12 POSITIVE EVEN INTEGERS. positive eve itegers: 2, 4, 6, 8,... = 12; a 1 = 2, d = 2 We use geeral term s formula to fid a 12, a a d a12 2 (12 1) 2 a Now, let's fid the sum: S a 1 2 a S 12 12(2 24) 2 156

15 S 2 a1 d( 1) 2 S is the sum of the first terms i a sequece is the umber of terms you are addig up a 1 is the first term of the sequece d is a commo differece

16 Fid the sum of the sequece: Fid S for a A theater has 60 seats i the first row, 68 seats i the secod row, 76 seats i the third row, ad so o i the same icreasig patter. If the theater has 20 rows of seats, how may seats are i the theater?

17 additio subtractio multiplicatio - (product) value eve umbers odd umbers sequece term of the sequece arithmetic progressio(sequece) recurrece formula commo differece d cosecutive series (sum of the terms) properties average a a 2 155=136+19

18 LESSON 2 A sequece is.. A commo umber commo to all cosecutive pairs of terms is.. Explai the recurrece formula.. A arithmetic sequece goes from oe term to the last by always addig (or subtractig) the same value. Is it correct? Read the formula 1 1 a A sum of the terms of a sequece is.. Write two formulas of the sum. a 2 a

19 Determie the value of b for which the sequece - a, -a/b, a/b, a is a arithmetic progressio... (a is ot 0) Which term of a arithmetic sequece 3,8,13 is 78? Is 150 a term of the series 11, 8, 5, 2,? The sum of first 3 term of decreasig arithmetic sequece is 9, the sum of squares of these terms is 99. Fid the fifth term of the sequece. Fid the fifth term of the arithmetic progressio, if ad S4 22 S8 92

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,