Sequences and Series
AP, GP, HP, AM-GM-HM inequalities, sums of special series.
Arithmetic and geometric progressions
nth term, sum of n terms, AM-GM inequality.
Arithmetic Progression (AP): each term differs from the previous by a constant d.
a, a + d, a + 2d, a + 3d, ...
- nth term: a_n = a + (n − 1)d
- Sum of first n terms: S_n = (n/2) · [2a + (n − 1)d] = (n/2)(a + a_n)
- Arithmetic mean (AM) of x and y: (x + y) / 2.
Worked example: Sum of integers from 1 to 100:
S = (100/2)(1 + 100) = 50 × 101 = 5050.
Geometric Progression (GP): each term is the previous multiplied by a constant r.
a, ar, ar², ar³, ...
- nth term: a_n = a · r^(n−1)
- Sum of first n terms: S_n = a(r^n − 1)/(r − 1) for r ≠ 1.
- Infinite sum (when |r| < 1): S_∞ = a / (1 − r).
- Geometric mean (GM) of x and y: √(xy).
Worked example: Sum of 1 + 2 + 4 + 8 + ... + 1024:
This is a GP with a=1, r=2, last term 1024 = 2¹⁰ → n=11 terms.
S = (2¹¹ − 1)/(2 − 1) = 2047.
Harmonic Progression (HP): sequence whose reciprocals form an AP.
- Harmonic mean (HM) of x and y: 2xy/(x + y).
AM-GM-HM inequality (for positive reals):
AM ≥ GM ≥ HM
Equality holds iff all terms are equal.
For two numbers: (x + y)/2 ≥ √(xy) ≥ 2xy/(x+y).
Useful relation: GM² = AM × HM.
Special series sums (for first n positive integers):
| Series | Sum |
|---|---|
| 1 + 2 + 3 + ... + n | n(n+1)/2 |
| 1² + 2² + 3² + ... + n² | n(n+1)(2n+1)/6 |
| 1³ + 2³ + 3³ + ... + n³ | [n(n+1)/2]² |
| 1 + 3 + 5 + ... + (2n−1) | n² |
| 2 + 4 + 6 + ... + 2n | n(n+1) |
Sum of cubes = (sum)² of the first n natural numbers.
Worked example: an AP-GP product series.
S = 1 + 2x + 3x² + 4x³ + ... (|x| < 1)
Standard trick: S − xS = 1 + x + x² + x³ + ... = 1/(1−x).
So (1 − x)S = 1/(1−x) → S = 1/(1−x)².
Special series sums
Σn, Σn², Σn³ formulas.