Sequence : A set of numbers arranged according to some definite rule is called a sequence.

The different numbers which form the sequence are called the terms of the sequence. Terms are denoted by the symbols T₁ T₂, T₃, … etc. the subscript denotes the position of the term. The nth term is also called the general term of the sequence.

Sequence containing finite number of terms is called a finite sequence and a sequence is called infinite, if it is not a finite sequence.

In a sequence, we should not always expect that the terms of a sequence will be necessarily given by an algebraic formula. For example, in a sequence of prime numbers 2, 3, 5, 7, …, there is no known formula for the prime numbers. But in the sequence, we do have a rule or law for writing its numbers in order.

Series : If the numbers forming the sequence are connected by the signs of addition (+), we get a series.
The numbers of the series are known as the terms of the series.

Progression : If the terms of a sequence constantly increase or decrease in a set pattern, it is called a progression.

Arithmetic Progression (A.P.) : A sequence in which the difference of any term from the previous term is constant, is called an arithmetic progression and the constant difference is called the common difference.

Notation :
(i) The first term is denoted by a.
(ii) The common difference is denoted by d.
(iii) d = T₂ – T₁ = T₃ – T₂ = T₄ — T₃ = … = T_n — T_n-1

nth term or general term of an A.P.: If a be the first term and d, the common difference of an A.P., then T_n = a + (n – 1 )d.
i.e., nth term = first term + (n – 1) x common difference.

Properties of an A.P.: 1. If a constant is added to each term of an A.P., the resulting sequence is also an A.P.

2. If a constant is subtracted from each term of an A.P., then resulting sequence is also an A.P.

3. If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.

4. If each term of an A.P. is divided by a non-zero constant, then the resulting sequence is also an A.P.

Sum to n terms of an A.P.
(i) S_n = \(\frac{n}{2}\) (a + 1), where T₁ = a, T_n = 1.

(ii) S_n = \(\frac{n}{2}\) [2a + (n – 1)d], where T₁, = a, c.d. = d.

Notes:
1. Formula (i) is used, when the last term is known and formula (ii) is used when the common difference is known.
2. These formulae have four quantities each. If three are known, the fourth can be found out.

Arithmetic Means : When three quantities are in A.P., the middle quantity is said to be Arithmetic Mean (A.M.) between the other two.

Thus, if, a, A, 6 are in A.P., then A is the A.M. between a and b.
∴ A-a = b-A [∵Each-c..d.]
⇒ 2A – a + b
A = \(\frac{a+b}{2}\)
A.M. between two numbers = Half their sum

An important result: Sum of n A.M.’s between two quantities is n times the single A.M. between them.
n Arithmetic means : Let A_1, A₂, A₃, …, A_n be the n A.M.’s between two numbers a and b.
The total number of A.M.’s and numbers a and b = (n + 2)
The last term = b.

b = a + (n + 2- 1 )d i.e., d = \(\frac{b-a}{n+1}\)
∴ We can find the A.M.’s as
A₁ = a + d = a + \(\frac{b-a}{n+1}\) , A₂ = a + 2d = a + \(2\frac{b-a}{n+1}\) ……
A_n = a + nd = a + \(n\frac{b-a}{n+1}\).

Geometric progression (G.P.): A sequence in which the ratio of any term to the proceeding term is the same throughout, is called a Geometric Progression and the constant ratio is called the common ratio of G.P.

Standard form of G.P. : If a is the first term and r is the common ratio, then a, ar, ar2, … is the standard form of G.P.

nth term or general term of G.P.:
T_n = ar^n-1>, where T₁ = a and c.r. = r.
The above formula has four quantities a, r, n and T_n. Out of these four, if any three are given, fourth can be found out.

Sum of n terms of a G.P.: If G.P. is a, ar, ar²,…; ar^n-1, then
(i) For, r = 1,
S_n = a + a + a + … + a (n terms)
= na
(ii) For r ≠ 1,
S_n = \(\frac{a\left(1-r^{n}\right)}{1-r}\), if |r| < 1
and S_n =\(\frac{a\left(r^{n}-1\right)}{1-r}\), if |r| > 1

Geometric Means : When three numbers are in G.P., then, middle one is called the Geometric Mean (G.M.) between the other two.

Thus, if a, G, b are in G.P., then G is the G.M. between a and b. It can also be defined as under :
If any number of numbers are in G.P., then all the terms between the first and last terms are called Geometric Means (G.M.’s) between them.

If a, G, b are in G.P. then
\(\frac{\mathrm{G}}{a}=\frac{b}{\mathrm{G}}\)
[v each = common ratio]
G² = ab ⇒ G = \(\sqrt{a b}\) .

An important result: The product of n G.M.’s between a and b is equal to the nth power of the G.M. between a and b.

n-Geometric Means : Let G₁ G₂, G₃, …, G_n be the n G.M.’s between a and b. Total number of terms = n G.M’s and the numbers a and b = n + 2.
∴ (n + 2)th term = last term = b.
If r is common ratio, b = ar^n+2-1 + 2 – 1 = arⁿ⁺¹
r = \(\left(\frac{b}{a}\right)^{n \frac{1}{n+1}}\)
G₁ = ar = a\(\left(\frac{b}{a}\right)^{\frac{1}{n+1}}\) ,
G₂ = ar² = a\(\left(\frac{b}{a}\right)^{\frac{2}{n+1}}\),……. G_n = arⁿ = a\(\left(\frac{b}{a}\right)^{\frac{n}{n+1}}\)

Sum of first n natural numbers :
Σn = 1 + 2 + 3 + … + n = \(\frac{n(n+1)}{2}\)

Sum of the squares of the first n natural numbers

Σn² = 1² + 2² + 3² + … + n² = \(\frac{n(n+1)(2 n+1)}{6}\)

Sum of the cubes of the first n natural numbers
Σn³ = 1³ + 2³ + 3³ + … + n³ = \(\left[\frac{n(n+1)}{2}\right]^{2}\)

Clearly, Σn³ = (Σn)²

Sum of it terms of a series whose nth term is an³ + bn² + cn + d :
If T_n = an³ + bn² + cn + d, then S_n = aΣn³ + bΣn² 4 cΣz + dn.