Sequences and Series Formula

Introduction

Sequence: A sequence is an arrangement of any objects or a set of numbers in a particular order followed by some rule. Example – 2, 4, 6, 8, 10, 12

• Finite sequence – A sequence containing a finite number of terms.
• Infinite sequence – A sequence containing an infinite number of terms.

Progression: A sequence whose terms follow a certain pattern is called a progression.

Series: If a sequence is expressed in sum, then it is called a series. Example – 2 + 4 + 6 + 10 + 12

Arithmetic Progression (AP)

A sequence in which terms increase or decrease regularly by a fixed number is called AP. This fixed number is called the common difference of AP.

Standard Form: a, a + d, a + 2d, …, a + (n-1)d

Where, a = first term and d = common difference

Formulae

1. General term: t_n = a + (n – 1)d
2. Sum of n term: S_n =  {n\over2}[2a + (n-1)d]
3. Sum of n term: S_n =  {n\over2}[a + l] l = last term of the sequence

Selection of Terms in an AP

1. Any three terms in AP can be taken as (a – d), a, (a + d)
2. Any Four terms in AP can be taken as (a – 3d), (a – d), (a + d), (a + 3d)
3. Any Four terms in AP can be taken as (a – 2d), (a – d), a, (a + d), (a + 2d)

Arithmetic Mean (AM)

If a, A, and b are in AP, then A is called the arithmetic mean of a and b and it is given by A = (a + b)/2

Geometric Progression (GP)

A sequence in which the ratio of any term (except the first term) to its just preceding term is constant throughout is called GP. This constant ratio is called the common ratio (r).

Standard Form: a, ar, ar², ar³ …., ar^n-1

Where, a = First term and

r = common ratio = 2^{nd}\ term\over 1^{st} term

Formulae

(i) General Term: t_n = ar^n-1

(ii) Sum of First n Term of GP:

• S_n = {a(1-r^n)\over1-r}, if r < 1
• S_n = {a(r^n-1)\over r-1}, if r > 1
• S_n = na, if = 1
• S_n = a-lr\over 1-r, r < 1 or S_n = lr-a\over r-1, r > 1

(iii) Sum of Infinite Terms of a GP:

• If |r| < 1, then S_∞ = a\over 1-r
• If |r| ≥ 1, then S_∞ = does not exist.

Geometric Mean (GM)

If a, G, and b are in GP, then G is called the geometric mean of a and b and it is given by G = √ab

Harmonic Mean (HM)

A sequence a₁, a₂, a₃, …, a_n of non-zero numbers is called a Harmonic Progression (HP), if the sequence {1\over a_1} + {1\over a_2}+{1\over a_3} +...+ {1\over a_n} is an AP.

General Term: t_n = {1\over a+(n-1)d}

Where, a = First Term and d = Common Difference

Harmonic Mean

If a, H, and b are in HP, then H is called the Harmonic mean of a and b and it is given by H = 2ab\over a+b

Properties of AM, GM, and HM between Two Numbers

If A, G and H are arithmetic, geometric and harmonic means of two positive numbers a and b, then

1. A = a+b\over 2, G = √ab, H = 2ab\over a+b
2. A ≥ G ≥ H
3. G² = AH and so A, G, H are in GP

Some Important Series

(i) Sum of first n natural number

1 + 2 + 3 + 4 + …. + n = n(n+1)\over 2

(ii) Sum of first n odd natural number

1 + 3 + 5 + 7 + …. + (2n – 1) = n²

(iii) Sum of first n even natural number

2 + 4 + 6 + 8 + … + 2n = n(n + 1)

(iv) Sum of square of first n natural number

1² + 2² + 3² + …. + n² = n(n+1)(2n+1)\over 6

(v) Sum of cube of first n natural number

1³ + 2³ + 3³ + …. + n³ = \Bigg[{n(n+1)\over 2}\Bigg]^2