**1. Sequence: **Sequence is a function whose domain is a subset of natural numbers. It represents the images of 1, 2, 3,… ,n, as f_{1}, f_{2}, f_{3}, …., f_{n} , where f_{n} = f(n).

**2. Real Sequence: **A sequence whose range is a subset of R is called a real sequence.

**3. Series: **If a_{1}, a_{2}, a_{3} , … , a_{n} is a sequence, then the expression a_{1} + a_{2} + a_{3} + … + a_{n} is a series.

**4. Progression: **A sequence whose terms follow certain rule is called a progression.

**5. Finite Series: **A series having finite number of terms is called finite series.

**6. Infinite Series: **A series having infinite number of terms is called infinite series.

**Arithmetic Progression (AP)**

A sequence in which the difference of two consecutive terms is constant, is called Arithmetic Progression (AP).

**Properties of Arithmetic Progression**

(i) If a sequence is an AP, then its nth term is a linear expression in n, i.e., its nth term is given by An + B, where A and B are constants and A = common difference.

(ii) nth Term of an AP If a is the first term, d is the common difference and / is the last term of an AP, then

(a) nth term is given by 1= a_{n} = a + (n – 1)d

(b) nth term of an AP from the last term is a’_{n} = l – (n – 1)d

(c) a_{n} + a’_{n} = a + 1

i.e., nth term from the start + nth term from the end

= constant

= first term + last term

(d) Common difference of an AP

d = T_{n} – T_{n-1}, ∀ n > 1

(e) T_{n} = 1/2[T_{n-k} + T_{n+k}], k < n

(iii) If a constant is added or subtracted from each term of an AP, then the resulting sequence is an AP with same common difference.

(iv) If each term of an AP is multiplied or divided by a non-zero constant k, then the resulting sequence is also an AP, with common difference kd or d/k where d = common difference.

(v) If a_{n}, a_{n+1} and a_{n+2} are three consecutive terms of an AP, then 2a_{n+1} = a_{n} + a_{n+2}.

(vi) (a) Any three terms of an AP can be taken as a – d, a, a + d.

(b) Any four terms of an AP can be taken as a-3d,a- d, a + d, a + 3d.

(c) Any five terms of an AP can be taken as a-2d,a – d, a, a + d, a + 2d.

(vii) **Sum of n Terms of an AP**

(a) Sum of n terms of AP, is given by S_{n} = n/2[2a + (n – 1)d] = n/2[a + l]

(b) A sequence is an AP, iff the sum of n terms is of the form An^{2} + Bn, where A and B are constants. Common difference in such case will be 2A.

(c) T_{n} = S_{n} – S_{n-1}

(viii) a^{2}, b^{2} and c^{2} are in AP.

(ix) If a_{1}, a_{2},…, a_{n} are the non-zero terms of an AP, then

**(x) Arithmetic Mean**

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

(b) If a_{1}, a_{2}, a_{3} , an are n numbers, then their AM is given by,

(c) If a, A_{1} , A_{2} , A_{3} ,…,A_{n}, b are in AP, then A_{1}, A_{2}, A_{3},…, A_{n} are n arithmetic mean between a and b, where

(d) Sum of n AM’s between a and b is nA

i.e., A_{1} + A_{2} + A_{3} + + = nA

**Geometric Progression (GP)**

A sequence in which the ratio of two consecutive terms is constant is called GP. The constant ratio is called common ratio (r).

i.e., a_{n+1}/a_{n} = r, ∀ n ≥ 1

**Properties of Geometric Progression (GP)**

(i) nth Term of a GP If a is the first term and r is the common ratio

(a) nth term of a GP from the beginning is a_{n} = ar^{n-1}

(b) nth term of a GP from the end is a’_{n} = l/r^{n-1}, l = last term

(c) If a is the first term and r is the common ratio of a GP, then the GP can be written as a, ar, ar^{2},… , ar^{n-1}, …

(d) The nth term from the end of a finite GP consisting of m terms is ar^{m-n}, where a is the first term and r is the common ratio of the GP.

(e) a_{n}a’_{n} = al i.e., nth term from the beginning x nth term from the end = constant = first term x last term.

(ii) If all the terms of GP be multiplied or divided by same non-zero constant, then the resulting sequence is a GP with the same common ratio.

(iii) The reciprocal terms of a given GP form a GP.

(iv) If each term of a GP be raised to same power, the resulting sequence also forms a GP.

(v) If the terms of a GP are chosen at regular intervals, then the resulting sequence is also a GP.

(vi) If a_{1}, a_{2}, a_{3}, … , a_{n} are non-zero, non-negative term of a GP, then

(a) GM = (a_{1}a_{2}a_{3}… a_{n} )^{1/n}

(b) log a_{1}, log a_{2}, log a_{3},…, log a_{n} are in an AP and vice-versa.

(vii) If a, b and c are three consecutive terms of a GP, then b^{2} = ac

(viii) (a) Three terms of a GP can be taken as a/r, a and ar.

(b) Four terms of a GP can be taken as a/r^{3}, a/r, ar and ar^{3}.

(c) Five terms of a GP can be taken as a/r^{2}, a/r, ar and ar^{2}.

**(ix) Sum of n Terms of a GP**

(a) Sum of n terms of a GP is given by

**(x) Geometric Mean (GM)**

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

(b) If a, G_{1}, G_{2}, G_{3}, , G_{n}, b are in GP, then G_{1}, G_{2}, G_{3},… , G_{n}, are in GM’s between a and b, where

(c) Product of n GM’s, G_{1} X G_{2} X G_{3} X … X G_{n} = G^{n}

**Harmonic Progression (HP)**

A sequence a_{1}, a_{2}, a_{3} ,…, a_{n} of non-zero numbers is called a Harmonic Progression (HP), if the sequence 1/a_{1}, 1/a_{2}, 1/a_{3}, …, 1/a_{n} is an AP.

**Properties of Harmonic Progression (HP)**

(i) nth term of HP, if a_{1}, a_{2}, a_{3} ,…, a_{n} are in HP, then

(a) nth term of the HP from the beginning

(b) nth term of the HP from the end

(d) a_{n} = 1/a+(n-1)d are the first term and common difference of the corresponding AP.

(ii) Sum of harmonic progression does not exist.

**Harmonic Mean**

(i) If a, H,b are in HP, then H is called the harmonic mean of a and b i.e., H= 2ab/(a + b)

(ii) If a, H_{1}, H_{2}, H_{3}, …, H_{n}, b are in HP, then

H_{1}, H_{2}, H_{3}, …, H_{n}

are n harmonic means between a and b where

(iii) Harmonic Mean (HM) between a_{1}, a_{2}, a_{3}, …, a_{n} is given by

**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

(i) A=(a+b)/2, G=√ab, H=(2ab)/(a+b)

(ii) A≥G≥H

(iii) A, G, H are in GP and G^{2} = AH

(iv) If A,G,H are AM, GM and HM between three given numbers a, b and c, then the equation on having a, b and c as its root is

(v) If A1,A2 be two AM’s, G_{1}, G_{2} be two GM’s and H_{1}, H_{2} be two HM’s between two numbers a and b, then

(vi) If A,G and H be AM, GM and HM between two numbers a and b, then

**Arithmetico-Geometric Progression**

A sequence in which every term is a product of a term of AP and GP is known as arithmetico-geometric progression.

The series may be written as

**Sum of Arithmetico-Geometric Series**

**Type 1** Let a_{l} + a_{2} + a_{3} + … be a given series. If a_{2} – a_{l}, a_{3} – a_{2}, … are in AP or GP, then a_{n} and S_{n} can be found by the method of difference.

where T_{1}, T_{2}, T_{3} ,… are terms of new series and S_{n} = Σa_{n}

**Type 2** It is not always necessary that the series of first order of differences i.e., a_{2} – a_{1}, a_{3} – a_{2}, …, a_{n} – a_{n-1} is always either in AP or in GP in such case.

Now, the series (T_{2} — T_{1}) + (T_{3} — T_{2}) + …+ (T_{n} – T_{n-1}) is series of second order of differences and when it is either in AP or in GP, then a_{n} = a_{1} + ΣT_{r}

Otherwise, in the similar way, we find series of higher order of differences and the nth term of the series.

**Exponential Series**

The sum of the series is denoted by thenumber e.

(i) e lies between 2 and 3.

(ii) e is an irrational number.

**Exponential Theorem**

Let a>0, then for all real value of x,

**Logarithmic Series**

**Important Result and Useful Series**

24. If number of terms in AP/GP/HP are odd, then AM/GM/HM of first and last term in middle term of progression.

25. If pth, qth and rth term of geometric progression are also in geometric progression.

26. If a,b and c are in AP and also in GP, then a=b=c

27. If a, b and c are in AP, then xa, xb and xc are in geometric progression.