**Introduction:**

**Sequence:** A sequence is a function whose domain is the set N of natural numbers.

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

In other words, a real sequence is a function with domain N and the range a subset of the set R of real numbers.

**Series:** If a₁, a₂, a₃, a₄ … a_{n} is a sequence, then the expression a₁ + a₂ + a₃ + a₄ + a₅ + … + a_{n} + … is a series.

**Progressions:** It is not necessary that the terms of a sequence always follow a certain pattern or they are described by some explicit formula for the nth term. Those sequences whose terms follow certain pattern are called progressions.

i) Every Progression is a series but every Series need not be a Progression.

ii) They are 4 types of Progressions (mainly 3)

1) Arithmetic Progression

2) Geometric Progression

3) Harmonic Progression

4) Arithmetic-Geometric Progression

**Arithmetic Progression (A.P):** A sequence is called an arithmetic progression if the difference of a term and the previous term is always same i.e. a_{n}₊₁ – a_{n} = constant (=d) for all n ϵ N. The constant difference, generally denoted by d is called the common difference.

**Illustration ****1:** 1, 4, 7, 10 … is an A.P. whose first term is 1 and the common difference is 4 – 1 = 3.

**Illustration 2: **11, 7, 3, -1 … is an A.P. whose first term is 11 and the common difference 7 – 11 = -4.

**Properties of an Arithmetic Progression:**

**Property I:** If a is the first term and d the common difference of an A.P., then its nth terms an is given by a_{n} = a + (n – 1) d.

**Property II:** A sequence is an A.P if its nth term is of the form case is An+ B i.e. a linear expression in n. The common difference is such a case is A i.e. the coefficient of n.

**Property III:** If a constant is added to or subtracted from each term of an A.P., then the resulting sequence is also an A.P. with the same common difference.

**Property IV:** If each term of a given A.P. is multiplied or divided by a non-zero constant k, then the resulting sequence is also an A.P. with common difference kd or d/k , where d is the common difference of the given AP.

**Property V:** In a finite A.P. the sum of the terms equidistant from the beginning and end is always same and is equal to the sum of first and last term i.e. a_{k} + a_{n}₋_{(k}₋₁_{)} = a₁ + a_{n}.

for all k = 1 ,2 ,3 ,…, n-1.

**Property VI:** Three numbers a, b, c are in A.P. if 2b = a + c.

**Property VII:** If the terms of an A.P. are chosen at regular intervals, then the form of an A.P.

**Property VIII:** If an, an+1 and an+2 three consecutive terms of an A.P., then, 2a_{n}₊₁ = a_{n} + a_{n}₊₂.

**Selection of Terms in an A.P**: The following ways of selecting terms are generally very convenient.

Number of Terms |
Terms | Common Difference |

3 |
a – d, a, a + d | d |

4 | a – 3d, a – d, a + d, a + 3d |
2d |

5 |
a – 2d, a – d, a, a + d, a + 2d |
d |

6 | a – 5d, a – 3d, a – d, a + d, a + 3d, a + 5d |
2d |

**Some Useful Results:**

\(\sum\limits_{r=1}^{n}{r}=1+2+…+n=\frac{n(n+1)}{2}\).

\(\sum\limits_{r=1}^{n}{{{r}^{2}}}={{1}^{2}}+{{2}^{2}}+…+{{n}^{2}}=\frac{n(n+1)\left( 2n+1 \right)}{6}\).

\(\sum\limits_{r=1}^{n}{{{r}^{3}}}={{1}^{3}}+{{2}^{3}}+…+{{n}^{3}}={{\left\{ \frac{n(n+1)}{6} \right\}}^{2}}\).

\(\sum\limits_{r=1}^{n}{{{r}^{4}}}={{1}^{4}}+{{2}^{4}}+…+{{n}^{4}}=\frac{n(n+1)(6{{n}^{3}}+9{{n}^{2}}+n-1)}{30}\).

**An Important Property: **A sequence is an A.P. if and only if the sum of its n terms is of the form An² + Bn. Where A, B are constants. In such a case, the common difference of the A.P is 2A.

**Remark: **It follows from this property that a sequence is an A.P. if the sum of its n terms is of the form An² + Bn. i.e., a quadratic expression in n and in such a case the common difference is twice the coefficient of n2. For example, if S_{n} = 3n² + 2n. We can say that it is sum of n terms of an A.P with common difference 6.

**Insertion of Arithmetic Means: **If between two given quantities a and b we have to insert n quantities A₁, A₂ … A_{n}. such that A₁, A₂ … A_{n}, b form an A.P. Then we say that A₁, A₂ … A_{n} are arithmetic means between a and b.

Example: Since 15, 11, 7, 3, -1, -5 are in A.P, it follows that 11, 7, 3, -1 are four arithmetic means between 15 and -5.

**Insertion of n Arithmetic Means between a and b:** Let A₁, A₂ … A_{n} be n arithmetic means between a and b. Then a, A₁, A₂ … A_{n}, b is an A.P. Let d be the common difference of this A.P. Clearly, it contains (n+2) terms.

∴ b = (n + 2)^{th} term

b = a + (n + 1)d

Now,

\({{A}_{1}}=\left( a+\frac{b-a}{n+1} \right)\).

A₂ = a + 2d ⇒ \({{A}_{2}}=\left( a+\frac{2\left( b-a \right)}{n+1} \right)…\).

A_{n} = a + nd ⇒ \({{A}_{n}}=\left( a+\frac{n\left( b-a \right)}{n+1} \right).\).

These are the required arithmetic means between a and b.

**Insertion of a Single Arithmetic Mean between a and b:** Let A be the arithmetic mean of a and b. Then a, A, b are in A.P.

A – a = b – A

2A = a + b

\(A=\frac{a+b}{2}\).