Arithmetic Progression Lecture Notes

Jul 16, 2024

Arithmetic Progression (AP)

Introduction to Arithmetic Progression

  • Sequence: 1, 4, 7, 10, 13, 16, 19
  • Arithmetic Progression: Sequence where the difference between consecutive terms is constant
    • Example: 4 - 1 = 3, 7 - 4 = 3, etc.
  • Common Difference (D): Difference between consecutive terms (D = 3 in above example)
  • First Term (A): Initial term of the series (A = 1 in above example)

Finding nth Term

  • Formula: Tn = A + (n-1)D
  • Example: Finding 6th term (T6)
    • T6 = 1 + (6-1) * 3 = 1 + 15 = 16

AP with Negative or Zero Differences

  • Decreasing AP Example: 5, 4, 3, 2, 1 (D = -1)
  • Constant AP Example: 3, 3, 3, 3 (D = 0)

Practice Questions

Question 1: Show Progression as AP

  • Given: √18, √50, √98
  • Simplified: 3√2, 5√2, 7√2
  • Common Difference: 2√2
  • Next Term: 7√2 + 2√2 = 9√2

Question 2: Find A and B in AP

  • Given: A, 9, B, 25
  • Forming Equations:
    • 9 - A = B - 9 = 25 - B
  • Solving gives A = 1 and B = 17

Question 3: Find 21st Term

  • Given AP: -5, -5/2, 0, 5/2...
  • D: 5/2
  • Formula: T21 = -5 + (21-1)*5/2 = -5 + 50 = 45

Question 4: Sum Conditions AP

  • Given: T4 + T8 = 24, T6 + T10 = 44
  • Form Equations and Solve for A and D:
    • A + 5D = 12, A + 7D = 22
    • Solving: A = -13, D = 5
  • First Three Terms: -13, -8, -3

Question 5: Find Middle Term

  • Given AP: -13, 205, 0, 5, 197..., up to 37
  • Calculating Number of Terms (N):
    • 37 = 213 + (n-1)*-8
    • n = 23
  • Middle Term: 12th term
    • T12 = 213 + 11*-8 = 125

Question 6: Prove AP Property

  • Given: mT_m = nT_n
  • Show: (m+n)th term = 0
  • Form Equations with AP formula and algebraic manipulation

Question 7: Eighth Term from End

  • Given AP: 7, 10, 13, ... 184
  • Two Methods:
    1. Formula: T_(N-n) = L - (n-1)D
    • T_8 = 184 - 21 = 163
    1. Reverse Series and Apply Normal Formula
    • Reverse AP: 184, 13, 10, 7
    • T8 = 184 + 7*-3 = 163

Question 8: Multiples of 4 Between 10-250

  • AP: 12, 16, 20,..., up to 248
  • N Calculation: 248 = 12 + (n-1)4
    • n = 60

Question 9: Selecting Terms in AP

  • 3 Terms: a - d, a, a + d
  • 4 Terms: a - 3d, a - d, a + d, a + 3d (Common Difference: 2d)
  • 5 Terms: a - 2d, a - d, a, a + d, a + 2d

Question 10: Finding AP with Product Condition

  • Equation Forming and Solving for given conditions
    • Given: Sum = 48, Product Condition
    • Answer: a = 16, d = 9
    • AP: 7, 16, 25

Additional Problem for Practice

  • Detailed Problem Given

Conclusion and Next Steps

  • AP concept and problem-solving summarized
  • Mention of upcoming videos on sum formulas