Exercise Set 3.2: Difference Tables and Sequences

Difference Table Exercises

Exercises 1 to 6: Construct a Difference Table to Predict the Next Term

• Sequence 1: 1, 7, 17, 31, 49, 71, …

  • First Differences: 6, 10, 14, 18, 22
  • Second Differences: 4, 4, 4, 4
  • Predicted Next Term: 71 + 26 = 97

• Sequence 2: 10, 10, 12, 16, 22, 30, …

  • First Differences: 0, 2, 4, 6, 8
  • Second Differences: 2, 2, 2, 2
  • Predicted Next Term: 30 + 10 = 40

• Sequence 3: 1, 4, 21, 56, 115, 204, …

  • First Differences: 3, 17, 35, 59, 89
  • Second Differences: 14, 18, 24, 30
  • Predicted Next Term: 204 + 118 = 322

• Sequence 4: 0, 10, 24, 56, 112, 190, …

  • First Differences: 10, 14, 32, 56, 78
  • Second Differences: 4, 18, 24, 22
  • Predicted Next Term: 190 + 122 = 312

• Sequence 5: 9, 4, 3, 12, 37, 84, …

  • First Differences: -5, -1, 9, 25, 47
  • Second Differences: 4, -8, 16, 22
  • Predicted Next Term: 84 + 62 = 146

• Sequence 6: 17, 15, 25, 53, 105, 187, …

  • First Differences: -2, 10, 28, 52, 82
  • Second Differences: 12, 18, 24, 30
  • Predicted Next Term: 187 + 114 = 301

nth-Term Formula Exercises

Exercises 7 to 10: Use Given nth-Term Formula to Compute First Five Terms

• Exercise 7: For the formula a_n = rac{n(2n + 1)}{2}

  • Compute:
  • a_1 = rac{1(2 imes 1 + 1)}{2} = 1
  • a_2 = rac{2(2 imes 2 + 1)}{2} = 6
  • a_3 = rac{3(2 imes 3 + 1)}{2} = 15
  • a_4 = rac{4(2 imes 4 + 1)}{2} = 28
  • a_5 = rac{5(2 imes 5 + 1)}{2} = 45

• Exercise 8: For the formula a_n = 3n

  • Compute:
  • a_1 = 3 imes 1 = 3
  • a_2 = 3 imes 2 = 6
  • a_3 = 3 imes 3 = 9
  • a_4 = 3 imes 4 = 12
  • a_5 = 3 imes 5 = 15

• Exercise 9: For the formula a_n = 5n^2 - 3n

  • Compute:
  • a_1 = 5 imes 1^2 - 3 imes 1 = 2
  • a_2 = 5 imes 2^2 - 3 imes 2 = 26
  • a_3 = 5 imes 3^2 - 3 imes 3 = 54
  • a_4 = 5 imes 4^2 - 3 imes 4 = 96
  • a_5 = 5 imes 5^2 - 3 imes 5 = 152

• Exercise 10: For the formula a_n = rac{2n^3}{n^2}

  • Compute:
  • Works out to be equal to 2n , thus:
  • a_1 = 2
  • a_2 = 4
  • a_3 = 6
  • a_4 = 8
  • a_5 = 10

Tiling Problems

Exercises 11 to 14: Determine nth-Term Formula for Square Tiles in nth Figure

• Exercise 11: Number of Tiles

  • Figures: 1, 4, 9, … leading to:
  • a_n = n^2

• Exercise 12: General term for Tiles

  • Consider observations leading to sequential series.

Cannonball Pyramids

Exercises 15 and 16: Cannonball Pyramids Sequence

• Patterns in Cannonballs:

  • Given values include:

  • a1 = 1, a2 = 4, a3 = 10, a4 = 20, a_5 = 35

  • Exercise 15a:

  • Differences: 3, 6, 10, 15

  • Use difference table:

    • Predict Sixth Pyramid: 56
    • Predict Seventh Pyramid: 84

  • Exercise 15b: Describe Eighth Pyramid

  • The expected count continues the pattern, leading to further geometric understanding of stacking pyramids.

• Exercise 16: Tetrahedral Sequence Formula

  • The nth-term formula for the tetrahedral sequence given as:
  • Tetrahedral_n = rac{1}{6} n(n + 1)(n + 2)
  • Finding Tetrahedral(10):
  • Calculation:
    • Tetrahedral_{10} = rac{1}{6} imes 10 imes 11 imes 12 = 220

Licorice Cuts

Exercises 17: Pieces vs. Cuts

• Understanding Cuts: One cut produces two pieces, two cuts produce three pieces, etc.

  • Exercise 17a: Count pieces:

  • ext{For 5 cuts: } 5 + 1 = 6 ext{ pieces}

  • ext{For 6 cuts: } 6 + 1 = 7 ext{ pieces}

  • Exercise 17b: Predict nth-term formula for Licorice Pieces:

  • Using observed patterns, it can be determined:

    • a_n = n + 1

Summary of Findings

• This study guide offers comprehensive exploration into difference tables, square tiling arrangements, cannonball pyramids, and licorice cut scenarios. Exploring each of these sequences and their nth-term formulas provides foundational understanding of mathematical sequences and their behavior respectively in both theoretical and practical applications.