Progressions — A.P., G.P. & H.P.

Sequence

• A sequence is an ordered list of numbers a₁, a₂, a₃, …, aₙ formed by a definite rule.

Arithmetic Progression (A.P.)

• Sequence with constant difference: d = aₙ − aₙ₋₁
• First term: a, Common difference: d
• n^th term: tₙ = a + (n − 1)d
• Sum of first n terms: Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l), where l = a + (n − 1)d
• Arithmetic Mean (A.M.) of a, b: (a + b)/2

Standard Sums

• Σn = n(n + 1)/2
• Σ(2n − 1) = n² (first n odd numbers)
• Σ(2n) = n(n + 1) (first n even numbers)
• Σn² = n(n + 1)(2n + 1)/6
• Σn³ = [n(n + 1)/2]²
• Σn⁴ = n(n + 1)(2n + 1)(3n² + 3n − 1)/30

Useful A.P. Properties

• If tₙ = an + b or Sₙ = an² + bn + c, the sequence is an A.P.
• Adding/subtracting a constant to each term keeps it an A.P.
• Multiplying/dividing each term by a non-zero constant keeps it an A.P.
• Sum of terms equidistant from start and end is constant (= first + last).
• If total terms are (2n + 1), sum = (2n + 1)aₙ₊₁.
• aₙ = Sₙ − Sₙ₋₁ (n ≥ 2)

Geometric Progression (G.P.)

• Sequence with constant ratio: r = aₙ / aₙ₋₁ (a₁ ≠ 0)
• n^th term: tₙ = arⁿ⁻¹
• Sum of first n terms: Sₙ = a(1 − rⁿ)/(1 − r) (r ≠ 1)
• Sum to infinity: S∞ = a/(1 − r) for |r| < 1
• Geometric Mean (G.M.) of a, b: √(ab)

Useful G.P. Properties

• Product of terms equidistant from ends is constant (= first × last).
• Multiplying/dividing all terms by a non-zero constant keeps it a G.P.
• If a G.P. has positive terms, then log a₁, log a₂, … form an A.P.

Harmonic Progression (H.P.)

• A sequence whose reciprocals form an A.P.
• n^th term of H.P. = reciprocal of n^th term of corresponding A.P.
• Harmonic Mean (H.M.) of a, b: H = 2ab/(a + b)

Relation between A.M., G.M. & H.M.

• G² = A·H
• A ≥ G ≥ H (equality when numbers are equal)

Insertion of Means

Arithmetic Means between a and b

• If n A.M.s are inserted, common difference: d = (b − a)/(n + 1)
• Sum of n A.M.s: n/2 (a + b)

Geometric Means between a and b

• If n G.M.s are inserted, common ratio: r = (b/a)^1/(n+1)
• Product of n G.M.s: (ab)^n/2

Arithmetico–Geometric Series

• General form: a + (a + d)r + (a + 2d)r² + …
• n^th term: tₙ = [a + (n − 1)d] rⁿ⁻¹
• Sum of infinite terms (|r| < 1): S = a/(1 − r) + dr/(1 − r)²

Difference Method

• If successive differences form an A.P., use A.P. summation on differences.
• If successive differences form a G.P., use G.P. summation on differences.