C Challenge

56. History When he was in the third grade, German mathematician Karl Gauss (1777–1855) took ten seconds to sum the integers from 1 to 100. Now it's your turn. Find a fast way to sum the integers from 1 to 100. Find a fast way to sum the integers from 1 to n. (Hint: Use patterns.)
57. Chess The small squares on a chessboard can be combined to form larger squares. For example, there are sixty-four 1 × 1 1 times 1  squares and one 8 × 8 8 times 8  square. Use inductive reasoning to determine how many 2 × 2 2 times 2  squares, 3 × 3 3 times 3  squares, and so on, are on a chessboard. What is the total number of squares on a chessboard?
58. 1. Algebra Write the first six terms of the sequence that starts with 1, and for which the difference between consecutive terms is first 2, and then 3, 4, 5, and 6.
    2. Evaluate n 2 + n 2 fraction n squared , plus n , over 2 end fraction  for n = 1 , 2 , 3 , 4 , 5 , n equals , 1 comma 2 comma 3 comma 4 comma 5 comma  and 6. Compare the sequence you get with your answer for part (a).

    3. Examine the diagram below and explain how it illustrates a value of n 2 + n 2 . fraction n squared , plus n , over 2 end fraction . .

       

    4. Draw a similar diagram to represent n 2 + n 2 fraction n squared , plus n , over 2 end fraction  for n = 5 . n equals , 5 .

Standardized Test Prep

SAT/ACT

59. What is the next term in the sequence 1, 1, 2, 3, 5, 8, 13,…?
    1. 17
    2. 20
    3. 21
    4. 24

60. A horse trainer wants to build three adjacent rectangular corrals as shown below. The area of each corral is 7200 ft 2 . 7200 , ft squared , .  If the length of each corral is 120 ft, how much fencing does the horse trainer need to buy in order to build the corrals?

    

    6. 300 ft
    7. 360 ft
    8. 560 ft
    9. 840 ft

Short Response

61. The coordinates x, y, a, and b are all positive integers. Could the points (x, y) and (a, b) have a midpoint in Quadrant III? Explain.

Mixed Review

See Lesson 1-8.

62. What is the area of a circle with radius 4 in.? Leave your answer in terms of π . pi .
63. What is the perimeter of a rectangle with side lengths 3 m and 7 m?

See Lesson 1-3.

64. Solve for x if B is the midpoint of A C ¯ . eh c bar , .  

Get Ready! To prepare for Lesson 2-2, do Exercises 65 and 66.

See Lesson 2-1.

Tell whether each conjecture is true or false. Explain.

65. The sum of two even numbers is even.
66. The sum of three odd numbers is odd.

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