SECTION 12.2 Arithmetic Sequences 869 Problems 111–119 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the subsequent sections, a final exam, or later courses such as calculus. Retain Your Knowledge 111. If $2500 is invested at 3% compounded monthly, find the amount that results after a period of 2 years. 112. Write the complex number − −i 1 in polar form. Express the argument in degrees. 113. For = − = + v i j w i j 2 and 2 , find the dot product ⋅ v w. 114. Find an equation of the parabola with vertex ( ) −3, 4 and focus ( ) 1, 4 . 115. Find the horizontal asymptote, if one exists, of ( ) = − − f x x x x 9 3 2 1 2 116. In a triangle, angle B is 4 degrees less than twice the measure of angle A, and angle C is 11 degrees less than three times the measure of angle B. Find the measure of each angle. 117. Find the average rate of change of ( ) = − y x tan sec 1 over the interval ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 10 3 , 10. 118. If ( ) = − + f x x x 5 2 9 2 and ( ) + = f a 1 16, find the possible values for a. 119. In calculus, the critical numbers for a function are numbers in the domain of f where ( ) ′ = f x 0 or ( ) ′ f x is undefined. Find the critical numbers for ( ) = − + − f x x x x 3 18 2 2 if ( ) ( ) ′ = − − − f x x x x 4 12 2 . 2 2 1. ( ) ( ) = = f f 2 1 2 ; 3 2 3 2. True 3. $1082.43 4. $9513.28 ‘Are You Prepared?’ Answers Explaining Concepts 109. Investigate various applications that lead to a Fibonacci sequence, such as in art, architecture, or financial markets.Write an essay on these applications. 110. Write a paragraph that explains why the numbers found in Problem 105 are called triangular. 1 Determine Whether a Sequence Is Arithmetic When the difference between successive terms of a sequence is always the same number, the sequence is called arithmetic . 12.2 Arithmetic Sequences OBJECTIVES 1 Determine Whether a Sequence Is Arithmetic (p. 869) 2 Find a Formula for an Arithmetic Sequence (p. 870) 3 Find the Sum of an Arithmetic Sequence (p. 871) DEFINITION Arithmetic Sequence An arithmetic sequence * is defined recursively as = − = − a a a a d , , n n 1 1 or as = = + − a a a a d n n 1 1 (1) where = a a 1 and d are real numbers. The number a is the first term, and the number d is called the common difference . Determining Whether a Sequence Is Arithmetic The sequence 4, 6, 8, 10, . . . is arithmetic since the difference of successive terms is 2. The first term is = a 4, 1 and the common difference is = d 2. EXAMPLE 1 *Sometimes called an arithmetic progression
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