53. Think About a Plan You invested money in a company and each month you receive a payment for your investment. Over the first four months, you received $50, $52, $56, and $62. If this pattern continues, how much do you receive in the tenth month?
    • What pattern do you see between consecutive terms?
    • Can you write a recursive or explicit formula to describe the pattern?
    • How can you use your formula to find the amount you receive in the tenth month?

54. Entertainment Suppose you are building towers of cards with levels as displayed below. Copy and complete the table, assuming the pattern continues.

    
    Image Long Description

    Number of Levels	Cards Needed
    1	2
    2	7
    3	□ white square
    4	□ white square
    5	□ white square

Find the next two terms in each sequence. Write a formula for the nth term. Identify each formula as explicit or recursive.

55. 5, 8, 11, 14, 17, …
56. 3, 6, 12, 24, 48, …
57. 1, 8, 27, 64, 125, …
58. 4, 16, 64, 256, 1024, …
59. 49, 64, 81, 100, 121, …
60. − 1 , 1 , − 1 , 1 , − 1 , 1 , … negative 1 comma 1 comma negative 1 comma 1 comma negative 1 comma 1 comma dot dot dot
61. − 16 , − 8 , − 4 , − 2 , … negative 16 comma negative 8 comma . negative 4 comma . negative 2 comma dot dot dot
62. − 75 , − 68 , − 61 , − 54 , … negative 75 comma negative 68 comma . negative 61 comma . negative 54 comma dot dot dot
63. 21 , 13 , 5 , − 3 , … 21 comma 13 comma 5 comma negative 3 comma dot dot dot
64. 1. Open-Ended Write four terms of a sequence of numbers that you can describe both recursively and explicitly.
    2. Write a recursive definition and an explicit formula for your sequence.
    3. Find the 20th term of the sequence by evaluating one of your formulas. Use the other formula to check your work.

65. Geometry Suppose you are stacking boxes in levels that form squares. The numbers of boxes in successive levels form a sequence. The figure below shows the top four levels as viewed from above.

    

    1. How many boxes of equal size would you need for the next lower level?
    2. How many boxes of equal size would you need to add three levels?
    3. Suppose you are stacking a total of 285 boxes. How many levels will you have?

C Challenge

66. Geometry The triangular numbers form a sequence. The diagram represents the first three triangular numbers: 1, 3, 6.

    

    1. Find the fourth and fifth triangular numbers.
    2. Write a recursive formula for the nth triangular number.
    3. Is the explicit formula a n = 1 2 ( n 2 + n ) eh sub n , equals , 1 half . open . n squared , plus n . close  the correct formula for this sequence? Explain.

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