Sequences and Series Formulas

A sequence is also referred to as a progression, which is defined as a successive arrangement of numbers in an order according to some specific rules. A series is formed by adding the elements of a sequence.

There are 3 main types of sequences, and the important formulas for each type are given below.

Arithmetic Progression (AP)

An arithmetic progression is a sequence in which each term differs from the previous one by a fixed amount, called the common difference (d).

• nth term: a_n = a + (n − 1)d
• Sum of n terms: S_n = n/2 · [2a + (n − 1)d] = n/2 · (a + l), where l is the last term
• Common difference: d = a_n − a_n-1
• Arithmetic mean of a and b: (a + b)/2
• If a, b, c are in AP: 2b = a + c
• a_n = S_n − S_n-1

Geometric Progression (GP)

In a geometric progression, each term is obtained by multiplying the previous one by a fixed number, the common ratio (r).

• nth term: a_n = a·r^n-1
• Sum of n terms: S_n = a(rⁿ − 1)/(r − 1) for r > 1, or a(1 − rⁿ)/(1 − r) for r < 1 (r ≠ 1); S_n = na if r = 1
• Sum to infinity (|r| < 1): S∞ = a/(1 − r)
• Common ratio: r = a_n/a_n-1
• Geometric mean of a and b: √(ab)
• If a, b, c are in GP: b² = ac

Harmonic Progression (HP)

A harmonic progression is defined through its reciprocals: a sequence is an HP if the reciprocals of its terms form an AP.

• Reciprocals of the terms form an AP
• nth term: a_n = 1 / [1/a + (n − 1)d]
• Harmonic mean of a and b: 2ab/(a + b)
• If a, b, c are in HP: b = 2ac/(a + c)

Note: Relation between AM, GM, HM

• AM ≥ GM ≥ HM
• (GM)² = AM × HM (for two numbers)

Infinite Series

• The sum of the terms of an infinite geometric series is given by,

S_n = a/(1−r)​

for |r| < 1, and not defined for |r| > 1