## Sequence and Series

**Sequence:** A sequence is an arrangement of number in a definite order, according to a definite rule.

**Terms:** Various numbers occurring in a sequence are called terms or element.

Consider the following lists of number:

3, 6, 9, 12, ……………

4, 8, 12, 16, ……………

-3, -2, -1, 0, ……………

In all the list above, we observe that each successive terms are obtained by adding a fixed number to the preceding terms. Such list of numbers is said to form on **Arithmetic Progression (AP).**

## Definition of Arithmetic Progression:

An **arithmetic progression** is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

This fixed number is called the common difference (d) of the **Arithmetic Progression**

Common difference can be positive, negative or zero.

Let us denote first term of A.P. by a or t, second term by a_{2} or t_{2} and nth term by a_{n} or t_{n} & the common difference by d. Then the **Arithmetic Progression** becomes

a_{1}, a_{2}, a_{3} …………………… a_{n}

where a_{2}

– a_{1} = d

or a_{2} = a_{1} + d

similarly a_{3} = a_{2} + d

Note:

In general, an – an-1 = d

or

an = an-1 + d

Thus

a, a + d, a + 2d, …………………..

forms an A.P. whose first term is ‘a’ & common difference is ‘d’

This is called general form of an A.P.

Finite A.P.: An A.P. containing finite number of terms is called finite A.P.

e.g. 147, 149, 151 ………………….. 163.

Infinite A.P.: An A.P. containing infinite terms is called infinite A.P.

e.g. 6, 9, 12, 15 ………………………..

## Problems Based on Sequence and Series

### Problem:

#### What is 18th term of the sequence defined by

### Solution:

We have,

Putting n = 18, we get

### Problem:

#### Let a sequence be defined by

#### ; . Find for n = 1, 2, 3, 4,

### Solution:

We have,

and,

Putting n = 3, 4, and 5, we have

a3 = a2 +a1 = 1 + 1 = 2

a4 = a3 +a2 = 2 + 1 = 3

a5 = a4 +a3 = 3 + 2 = 5

\ we have

a1 = 1, a2 = 1, a3 = 2, a4 = 3 and a5 = 5

Now, putting n = 1, 2, 3 and 4 in we have

[\a1 = a2 = 1]

[\a2 = 1 and a3 = 2]

[\a3 = 2 and a4 = 3]

[\a4 = 3 and a5 = 5]

### Problem:

#### Which of the following are A.P’s? If they form an A.P., find the common difference d and the next three terms after last given term.

#### (i) 2, 4, 8, 16, … (ii) …

#### (iii) … (iv) 12, 52, 72, 73, …

### Solution:

(i)

2, 4, 8, 16, …

As the difference between two Consecutive terms is not the same

\ The given sequence is not an A.P.

(ii)

As the common difference between any two consecutive terms is the same

\ The given sequence is an A.P.

Next three terms are :

(iii)

As the common difference between any two consecutive terms is not the same

\ The given sequence is not an A.P.

(iv)

As the common difference between any two consecutive terms is the same

\ The given sequence is an A.P.

Next three terms are :

### Problem:

#### Write the A.P., when the first term a and common difference d is as follows:

#### (i) a = 10, d = 10 (ii) (iii)

### Solution:

(i) a = 10, d = 10

\ The required A.P. is:

10, 10 + 10, 10 + 2 ´ 10, 10 + 3 ´ 10, …

or 10, 20, 30, 40, …

(ii)

\ The required A.P. is:

or

(iii)

\ The required A.P. is:

or

### Problem:

#### For the following A.P.’s write the first term and the common difference:

#### (i) (ii) (iii)

### Solution:

(i)

First term:

(ii)

First term:

(iii)

First term:

### Problem:

#### Which of the following list of numbers will form an AP?

#### (i) The amount of air present in the cylinder when a vacuum pump removes each time th of air remaining in the cylinder.

#### (ii) The cost of digging a well for the first meter is Rs. 150 and rises by Rs. 20 for each succeeding meter.

### Solution:

(i)

Let the initial volume = V

Air removed in 1st stage

Vol. of air left at stage

Air removed in 2nd stage

Vol. of air left at stage

Air removed in 3rd stage =

Vol. of air left at stage

Now, the difference between the volumes of air left in the initial stage and stage 1 is given by

Similarly, the difference between the volumes of air left in the stage 1 and stage 2 is given by

Also, such difference between the stage 2 and stage 3 is given by

Since the difference in the volumes at the first three stages are

which are not equal, it is not an A.P.

(ii) Since the cost of digging a well rises by Rs. 20

\ the cost of digging well becomes:

150 + (150 + 20) + (150 + 40) + ………..

i.e., 150 + 170 + 190 + ……………

\ The above series forms an A.P.