Sequence and Series Formula Sheet

Arithmetic Progression (AP)

General term: an = a + (n-1)d

Sum of n terms: Sn = n/2 [2a + (n-1)d] = n/2 (a1 + an)

Shortcuts:

  1. If Tm = n and Tn = m, then T(m+n) = 0
  2. If Tm = n and Tn = m, then Tr = m+n-r
  3. If n * Tn = m * Tm, then T(m+n) = 0

Geometric Progression (GP)

General term: an = ar(n-1)

Sum of n terms: Sn = a(1-rn)/(1-r) for r ≠ 1

Sum of infinite terms: S∞ = a/(1-r) for |r| < 1

Arithmetic-Geometric Progression (AGP)

General term: an = (a + (n-1)d)r(n-1)

Sum of n terms: Sn = (a/d - r(a-d)/(d(1-r)))(1-rn) + (a-d)n/d

Harmonic Progression (HP)

If a, b, c are in HP, then 1/b = (1/a + 1/c)/2

To insert n HMs between a and b: Hk = 2ab/[(a+b) + (b-a)(2k-1)/(n+1)]

Means

Arithmetic Mean (AM)

AM = (a + b)/2

To insert n AMs between a and b: AMk = a + k(b-a)/(n+1)

Geometric Mean (GM)

GM = √(ab)

To insert n GMs between a and b: GMk = a(b/a)k/(n+1)

Harmonic Mean (HM)

HM = 2ab/(a+b)

Relationship

For positive numbers: AM ≥ GM ≥ HM

AM, GM, HM are in GP: GM2 = AM * HM

Miscellaneous Series

  1. Sum of squares: Σn2 = n(n+1)(2n+1)/6
  2. Sum of cubes: Σn3 = (n(n+1)/2)2
  3. S = 1·2 + 2·3 + 3·4 + ... + n(n+1) = n(n+1)(n+2)/3
  4. S = 1·2·3 + 2·3·4 + 3·4·5 + ... + n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4
  5. For series like an = n2 + bn + c:
    Sn = (n/6)[2n2 + 3n + 1]a + (n/2)[n + 1]b + nc

Special Results

  1. Minimum value of xα + x(-α) where α > 0: 2√α
  2. Minimum value of 2(sinx) + 2(cosx): 2(1-1/√2)