📌 Key Points
- Arithmetic Progression (AP): Sequence with constant difference between consecutive terms
- Common difference d = aₙ₊₁ - aₙ (difference between any two consecutive terms)
- General term formula: aₙ = a + (n-1)d, where a is first term
- Sum of n terms: Sₙ = n/2 × [2a + (n-1)d] or Sₙ = n/2 × [a + l], where l is last term
- Arithmetic Mean of two numbers a and b: A = (a + b) / 2
- Inserting n arithmetic means between a and b: total terms = n+2, d = (b-a)/(n+1)
- In AP with odd number of terms, middle term = (first + last) / 2
- Sum of first n natural numbers: Sₙ = n(n+1)/2
- Sum of first n odd numbers: Sₙ = n²
- Sum of first n even numbers: Sₙ = n(n+1)
- If Sₙ = An² + Bn, then a = S₁ and d can be found from S₂ - S₁
- To check if sequence is AP: differences between consecutive terms must be equal
- For solving problems with two conditions (like 4th term and 7th term), use two equations
- Common difference can be positive (increasing AP), negative (decreasing AP), or zero
- AP is defined by any two of: first term a, common difference d, nth term aₙ, sum Sₙ
- Word problems: identify first value (a) and uniform change (d), then apply formulas
- To find if a number is in AP: set aₙ = number and check if n is positive integer
- Never confuse AP with geometric progression (constant ratio) or other sequences
- For AP starting from different values: carefully identify what is a (first term)
- Always verify answers make sense: check specific terms, verify common difference
📘 Important Definitions
🔢 Formulas & Laws
General Term
aₙ = a + (n-1)d
Sum of n Terms (Form 1)
Sₙ = n/2 × [2a + (n-1)d]
Sum of n Terms (Form 2)
Sₙ = n/2 × [a + l], where l = a + (n-1)d is the last term
Arithmetic Mean
A = (a + b) / 2
Common Difference
d = aₙ - aₙ₋₁ (difference between consecutive terms)
Inserting n Means
d = (b - a) / (n + 1) for inserting n means between a and b
Sum of Natural Numbers
1 + 2 + 3 + ... + n = n(n+1)/2
Sum of Odd Numbers
1 + 3 + 5 + ... + (2n-1) = n²
⚠️ Common Mistakes
✗ Wrong: Using aₙ = a + nd instead of aₙ = a + (n-1)d
✓ Correct: There are (n-1) gaps between first and nth term. Always use (n-1)d, not nd.
✗ Wrong: Forgetting to divide by 2 in sum formula
✓ Correct: Sₙ = n/2 × [...], not n × [...]. The division by 2 is essential.
✗ Wrong: Confusing d = (b-a)/n with d = (b-a)/(n+1) when inserting means
✓ Correct: When inserting n means between a and b, there are (n+1) gaps, so d = (b-a)/(n+1).
✗ Wrong: Not checking if common difference is constant
✓ Correct: Always verify d by calculating differences between at least 3 pairs of consecutive terms.
✗ Wrong: Ignoring negative common differences
✓ Correct: If d < 0, the AP is decreasing. This is valid. Example: 10, 5, 0, -5, ... has d = -5.
✗ Wrong: Wrong identification of first term in word problems
✓ Correct: First term a is the starting value. If problem starts from 'month 1' or 'year 1', use that value as a.
✗ Wrong: Confusing AP with other sequences (geometric, arithmetic-geometric, etc.)
✓ Correct: AP has constant difference. Geometric has constant ratio. Always check what type it is.
✗ Wrong: Not verifying if a number is in AP by checking if n is positive integer
✓ Correct: To check if k is in AP: solve aₙ = k. If n is positive integer, k is in AP. Otherwise, it's not.
📝 Exam Focus
These questions are frequently asked in CBSE exams:
🎯 Last-Minute Recall
Close your eyes and try to recall: Key definitions, formulas, and 3 common mistakes. If you can recall 80% without looking, you're exam-ready!