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
  2. 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