1. Definition of AP
An Arithmetic Progression (AP) is a sequence where:
Difference between consecutive terms is constant
This constant difference is called common difference (d)
- First term: a1 or simply 'a'
- Common difference: d = a2 - a1 = a3 - a2 = ...
- General AP: a, a+d, a+2d, a+3d, ...
2. nth Term of AP
a? = a + (n-1)d
Where:
- a? = nth term
- a = first term
- d = common difference
- n = term number
Important Relations:
d = (a? - a)/(n-1)
n = [(a? - a)/d] + 1
To check if a number is in AP: (middle term) = (sum of neighbors)/2
3. Sum of First n Terms of AP
S? = n/2 [2a + (n-1)d]
OR S? = n/2 [a + a?]
Where:
- S? = Sum of first n terms
- n = number of terms
- a = first term
- a? = last term (nth term)
4. Properties of AP
Constant Difference
a??1 - a? = d (constant)
Three Terms in AP
a-d, a, a+d
Four Terms in AP
a-3d, a-d, a+d, a+3d
5. Finding Terms in AP
If sum and product of terms are given
- Let terms be a-d, a, a+d
- Use given conditions to form equations
- Solve for a and d
- Find all terms
Inserting Arithmetic Means
To insert k arithmetic means between a and b:
Total terms = k + 2
Common difference d = (b-a)/(k+1)
AP: a, a+d, a+2d, ..., a+(k+1)d = b
6. Word Problem Types (NCERT)
- Saving money problems: Monthly savings in AP
- Salary increment problems: Yearly increments
- Loan repayment: Installments in AP
- Seating arrangement: Rows in AP
- Number of trees/plants: In AP pattern
- Competition prizes: Decreasing amounts
7. Important Formulas
Sum of first n natural numbers: n(n+1)/2
Sum of squares of first n natural numbers: n(n+1)(2n+1)/6
Sum of cubes of first n natural numbers: [n(n+1)/2]²
If S? = An² + Bn, then d = 2A and a = A + B
8. Quick Reference Table
General Form
a, a+d, a+2d, ...
Sum of n terms
S? = n/2[2a+(n-1)d]
9. Special Cases
Finite AP
AP with last term l: a, a+d, a+2d, ..., l
Number of terms: n = (l-a)/d + 1
Sum: S? = n/2(a+l)
Negative Common Difference
When d < 0, AP is decreasing
Example: 10, 7, 4, 1, -2, ... (d = -3)
10. Example (NCERT Style)
Problem: Find 10th term of AP: 2, 7, 12, ...
Solution:
a = 2, d = 7-2 = 5
Using a? = a + (n-1)d
a10 = 2 + (10-1)Χ5
= 2 + 9Χ5 = 2 + 45 = 47
Problem: Find sum of first 20 terms of above AP
Solution:
S20 = 20/2 [2Χ2 + (20-1)Χ5]
= 10 [4 + 19Χ5] = 10 [4 + 95] = 10 Χ 99 = 990