## Sequence:

A set of numbers arranged in a definite order according to some definite rule is called a **sequence**. Each number of the set is called a **term** of the sequence.

**Finite and Infinite sequence: **

A sequence is called finite or infinite according to the number of terms in it is finite or infinite.

**General term of sequence: **

Let us consider the sequence of "cubes" of natural numbers: 1, 8, 27, 64, . . . .

The different terms of a sequence are usually denoted by t_{1}, t_{2}, t_{3}, . . . etc.

Here, the subscript (always a natural number) denotes the position of the term in the sequence.

Thus, in the above sequence t_{1} = 1; t_{2} = 8; t_{3} = 27 . . . . . etc.

Hence, First term = t_{1} = 1, Second term = t_{2} = 8, Third term = t_{3} = 27, . . . etc.

In general, n^{th} term = t_{n}, which is called **general term** of the sequence.

Often, it is possible to express the rule which generates the various terms of the sequence in terms of an algebraic formula.

In the above sequence 1, 8, 27, 64, 125 . . ., n^{th} term = t_{n} = n^{3}.

Thus, the rule for the above sequence is n^{3}, where n is any natural number.

## Series:

If a_{1}, a_{2}, . . ., a_{n} is a sequence of numbers,

then the expression a_{1} + a_{2} + . . . + a_{n}

is called **series** associated with the given sequence.

Like a sequence, a series may also be finite or infinite.

It is common to represent a series compactly using the **Σ** (sigma) symbol. Sigma indicates a summation as:

a_{i} = a_{1} + a_{2} + a_{3} + . . . + a_{n} for a finite series.

Note that 'i' takes the values from '1' to the number indicated at the top of the sigma symbol.

(The L.H.S. is read as "sigma a_{i}, i equal to 1 to n")

a_{i} = a_{1} + a_{2} + a_{3} + . . . + a_{n} + . . . ∞ for an infinite series.

An arithmetic progression is a sequence in which each term (except the first term) is obtained by adding a fixed number (positive or negative or zero) to the term immediately preceding it.

Hence, this fixed number becomes the difference of two successive terms.

For this reason it is called as the **common difference ** and is usually denoted by **d**.

Quantities are said to be in Arithmetic Progression (A.P.) when they increase or decrease by a common difference. The common difference is formed by subtracting any term of the sequence from that which follows it.

Thus, if t_{1}, t_{2}, t_{3}, . . . . . , t_{n} are the terms in an A.P. and the common difference is 'd', then |

t_{2} = t_{1} + d d = t_{2} – t_{1} t _{3} = t_{2} + d d = t_{3} – t_{2} t _{4} = t_{3} + d d = t_{4} – t_{3} |

- - - - - - - - - - - - - - - - t _{n} = t_{n - 1} + d d = t _{n} – t_{n - 1} |

In above, if the first term t, then
_{1} = a |

t_{2} = t_{1} + d = a + d = a + (2 – 1)d |

t_{3} = t_{2} + d = [a + d] + d = a + 2d = a + (3 – 1)d |

t_{4} = t_{3} + d = [a + 2d] + d = a + 3d = a + (4 – 1)d |

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |

t_{n} = t_{n – 1 } + d = a + (n – 1)d t is called _{n}general term of A.P. Thus t
_{n} = a + (n – 1)d |

By substituting n = 1, 2, 3, . . . we get a, a + d, a + 2d, a + 3d, a + 4d, . . . . . . . . It represents an arithmetic progression where "a" is the first term and "d" is the common difference. This is called the general form of an A.P. The common difference (d) is given by t _{n} – t_{n – 1}. |

When three quantities are in A.P., the middle one is said to be the **arithmetic mean** of the other two.

Thus, if a, b, c are three successive quantities in A.P., then "b" is called the arithmetic mean of a and c.

It is given by **b = **

Hence, arithmetic mean of two numbers is half of their sum i.e, their average.

(It may be noted that, while the AM used in A.P. is average of just two numbers, the term **mean ** used in statistics is the average of entire set of data).

## Properties of Arithmetic Progression:

1. If a, b, c are in A.P., then **2b = a + c**

2. If a, b, c are in A.P. and k ≠ 0, then the following are also in A.P.

(i) a + k, b + k, c + k i.e, when k is added to each term

(ii) a – k, b – k, c – k i.e, when k is subtracted from each term

(iii) ak, bk, ck i.e, when each term is multiplied by k

(iv) a/k, b/k, c/k i.e, when each term is divided by k

Refer to an actual incident explained at the right.

We will now use the same technique to find the sum of the first "n" terms of an A.P.

Let the first term of an A.P. be **a** and common difference be **d**.

Let **S _{n}** denote the sum of the n terms of A.P. Then

S

_{n}= a + [a + d] + [a + 2d] + . . . . + [a + (n – 2)d] + [a + (n – 1)d]

Rewriting the terms in reverse order, we have

S

_{n}= [a + (n – 1)d] + [a + (n – 2)d] + . . . . + [a + 2d] + [a + d] + a

On adding the two

S

_{n}+ S

_{n}= [2a + (n – 1)d] + [2a + (n – 1)d] + . . . . . . .+ [2a + (n – 1)d] +

[2a + (n – 1)d] (n times)

⇒ 2S

_{n}= n[2a + (n – 1)d]

⇒

**S**

_{n}**=**

**[2a + (n – 1)d]**

We can also write

S

_{n}= [a + a + (n – 1)d] = [a + a

_{n}] since a + (n – 1)d = a

_{n}

Sometimes S

_{n}is simply denoted by S.

If there are only 'n' terms in an A.P. and a

_{n}= l (the last term), then

**S = [a + l]**

The above formula is useful when the first and last terms of the A.P. are known.

In solving problems related to Arithmetic progressions, the following shall be useful:

**i.** 3 successive terms of an A.P. can be considered as: ** (a – d), a, (a + d)**

**ii.** 4 successive terms of an A.P. can be considered as: ** (a – 3d), (a – d), (a + d), (a + 3d)**

**iii.** 5 successive terms of an A.P. can be taken as: ** (a – 2d), (a – d), a, (a + d), (a + 2d) **

**iv.** The n^{th} term of an A.P. is equal to sum of n terms minus sum of (n – 1) terms

i.e, **T _{n} = S_{n} – S_{n-1}**

**v.**The sum of terms equidistant from beginning and end is constant

An Arithmetic progression involves the basic principle of counting.

**a.** If you deposit the same amount of money every week in a piggy bank, the weekly total amounts form an A.P.

**b.** In an aluminum ladder with sloping sides, the lengths of each rung would form an A.P. (See adjacent figure)

**c.** If you hire an auto or a taxi, you will be charged a fixed amount (say Rs. x) for a certain distance. Thereafter, you will have to pay additional charges per every km travelled (say Rs. y).

So, the total fares at the end of every km form an A.P. as:

x, x + y, x + 2y, x + 3y, etc