Arithmetic Progression (A.P.)
aₙ = a + (n−1)d
Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l) (l = last term)
Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l) (l = last term)
Key Properties
- a₁ + aₙ = a₂ + aₙ₋₁ = a₃ + aₙ₋₂ = constant
- If tₙ is linear in n → sequence is A.P. (d = coefficient of n)
- If Sₙ is quadratic in n with constant term 0 → A.P. (d = 2 × coeff of n²)
- If {aₙ} is A.P. with aₙ ≠ 0 → {1/aₙ} is G.P.
Common Difference
d = (tₚ − t_q)/(p − q) (p, q ∈ N)
Geometric Progression (G.P.)
aₙ = a rⁿ⁻¹
Sₙ = a(rⁿ − 1)/(r − 1) (r ≠ 1)
Sₙ = n a (r = 1)
Sum to infinity (|r| < 1): S∞ = a/(1 − r)
Sₙ = a(rⁿ − 1)/(r − 1) (r ≠ 1)
Sₙ = n a (r = 1)
Sum to infinity (|r| < 1): S∞ = a/(1 − r)
Key Properties
- a₁ aₙ = a₂ aₙ₋₁ = a₃ aₙ₋₂ = constant (square of middle term)
- If {aₙ} is G.P. (aₙ > 0) → {log aₙ} is A.P. (converse true)
Arithmetic Mean & Geometric Mean
Arithmetic Mean (A.M.)
Between a & c: A = (a + c)/2
For n numbers: A = (a₁ + a₂ + ... + aₙ)/n
Geometric Mean (G.M.)
Between a & c: G = √(a c)
For n positive numbers: G = (a₁ a₂ ... aₙ)1/n
A.M. ≥ G.M. (Equality iff all equal)
Sum of Special Series
Sum of first n natural numbers
S = n(n+1)/2
Sum of squares
1² + 2² + ... + n² = n(n+1)(2n+1)/6
Sum of cubes
[n(n+1)/2]²
Important Properties & Results
- Three numbers a, b, c in A.P. ⇔ 2b = a + c
- Three numbers a, b, c in G.P. ⇔ b² = a c
- If a, b, c, d in A.P. → b − a = c − b = d − c (common difference)
- Infinite G.P. sum exists only when |r| < 1
- Arithmetic mean of two numbers ≥ their geometric mean
- If A.P. and G.P. have same first & last term → A.M. > G.M. unless a = l
JEE Main Tips & Weightage
Highest weightage chapter in Algebra — 5–7 questions every year!
Must Know for JEE
- nᵗʰ term & sum formulae of A.P. & G.P.
- Infinite G.P. sum = a/(1−r) (|r|<1)
- Sum of n natural numbers, squares, cubes
- Inserting n A.M.s or G.M.s between a & b
- A.P. ↔ G.P. conversions (reciprocals, logs)
- Three terms in A.P./G.P. problems
- Σ sign change problems