SECTION 12.3 Geometric Sequences; Geometric Series 875 70. Citrus Ladders Ladders used by fruit pickers are typically tapered with a wide bottom for stability and a narrow top for ease of picking. If the bottom rung of such a ladder is 49 inches wide and the top rung is 24 inches wide, how many rungs does the ladder have if each rung is 2.5 inches shorter than the one below it? How much material would be needed to make the rungs for the ladder described? Source: www.stokesladders.com 71. Challenge Problem If { } an is an arithmetic sequence with 100 terms where = a 2 1 and = a 9, 2 and { } bn is an arithmetic sequence with 100 terms where = b 5 1 and = b 11, 2 how many terms are the same in each sequence? 72. Challenge Problem Suppose { } an is an arithmetic sequence. If Sn is the sum of the first n terms of { } a , n and S S n n 2 is a positive constant for all n, find an expression for the n th term, a ,n in terms of only n and the common difference, d. Retain Your Knowledge Problems 75–84 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 subsequent sections, a final exam, or later courses such as calculus. 75. If a credit card charges 15.3% interest compounded monthly, find the effective rate of interest. 76. The vector v has initial point ( ) = − P 1, 2 and terminal point ( ) = − Q 3, 4 . Write v in the form + a b i j; that is, find its position vector. 77. Analyze and graph the equation: + = x y 25 4 100 2 2 78. Find the inverse of the matrix − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 2 0 3 1 , if it exists; otherwise, state that the matrix is singular. 79. Find the partial fraction decomposition of − x x 3 1 . 3 80. Find the exact value of π π − sin 5 8 cos 5 8 . 2 2 81. If g is a function with domain [ ] −4,10 , what is the domain of the function ( ) − g x 2 1 ? 82. Find the real zeros of ( ) ( ) ( ) ( ) = + ⋅ − − ⋅ + h x x x x x x 1 2 1 4 1 4 2 3 4 83. Identify the curve given by the equation ( ) + + − = y x y x 6 24 12 0 2 2 84. Solve: ( ) ( )( ) + = + − + x x x 3 3 5 7 2 Explaining Concepts 73. Make up an arithmetic sequence. Give it to a friend and ask for its 20th term. 74. Describe the similarities and differences between arithmetic sequences and linear functions. OBJECTIVES 1 Determine Whether a Sequence Is Geometric (p. 875) 2 Find a Formula for a Geometric Sequence (p. 877) 3 Find the Sum of a Geometric Sequence (p. 878) 4 Determine Whether a Geometric Series Converges or Diverges (p. 879) 5 Solve Annuity Problems Using Formulas (p. 881) 12.3 Geometric Sequences; Geometric Series Now Work the ‘Are You Prepared?’ problem on page 883. • Compound Interest (Section 5.7, pp. 345–351) PREPARING FOR THIS SECTION Before getting started, review the following: 1 Determine Whether a Sequence Is Geometric When the ratio of successive terms of a sequence is always the same nonzero number, the sequence is called geometric .
RkJQdWJsaXNoZXIy NjM5ODQ=