A ** sequence** is a set of numbers either having a theoretical scheme ( verbal description) or having a general formula for the nth

**term**. A finite sequence is a sequence with finite

**terms**in it whereas, an infinite sequence is a sequence with infinitely many

**terms**. A

**is the sum of the terms of a sequence, which has a definite formula for it’s nth term.**

*series*Mathematically, if a

_{1}, a

_{2}, a

_{3}, … , a

_{n}is a sequence, then the series associated with it is: a

_{1}+ a

_{2}+ a

_{3}+ … + a

_{n}.

A series is finite or infinite according to the sequence it is associated with.

## What is a term of a sequence?

The various numbers of a sequence are called its terms. The nth term is the number at the nth position and is denoted by a_{n}. The nth term is also called the general term of sequence.

## Different types of sequences

Depending on the formula for the general term, we have different types of sequences. For example, some Common sequences are:

- Arithmetic sequence
- Geometric sequence
- Harmonic sequence
- Fibonacci sequence*

## Arithmetic Sequences

If any two successive terms of a sequence differ by a constant number, known as *common difference*, then the sequence is an arithmetic sequence.

For example:

i) 2, 5, 8 ,11, 14, 17

ii) 10, 5, 0, -5, -10, -15

iii) Set of Natural numbers

iv) Set of Whole numbers

v) Set of Even numbers

## Geometric Sequences

If the ratio of any two successive terms of a sequence is constant, then the sequence is a geometric sequence. Here, the ratio of a_{n+1} to a_{n} is known as *common ratio*.

For example:

i) 2, 1, 1/2, 1/4, 1/8.

ii) 3, 6, 12, 24, 48.

iii) 5, 25, 125, 625, 3125.

## Harmonic Sequences

If the inverses of any two successive terms of a sequence is constant, then the sequence is a harmonic sequence. Thus, the inverses of terms of a harmonic sequence form an arithmetic sequence, by definition.

Ex:

i) 1/4, 1/8, 1/12, 1/16, 1/20.

ii) 1, 1/2, 1/3, 1/4, 1/5, 1/6.

## Fibonacci Sequence*

Each term of the sequence is the sum of the two preceding terms, with the first two terms fixed as 0, 1.

For instance, the first 10 terms of a fibonacci sequence are 0, 1, 1, 2 3, 5, 8, 13, 21, 34.

## Solved problems on sequence and series

** Example 1**: If 3, 5, 7, 9, … ,53 is an arithmetic sequence, then find the common difference and the 21st term in the sequence?

** Solution**:

Common difference of an Arithmetic sequence is the difference between any two successive terms of the sequence.

Taking the first two terms. a

_{2}= 5, a

_{1}= 3,

Common difference = d = a

_{n}– a

_{n-1}

= a

_{2}– a

_{1}= 5 – 3

d = 2.

Formula for the n

^{th}term of arithmetic sequence = a

_{1}+ (n – 1) d

For n = 21, we have a

_{21}= 3 + (21 – 1) * 2

= 3+ 40 = 43

21st term of the sequence = a

_{21}= 43.

** Example 2:** Find the sum of the series 3 + 6 + 9 + … + 42.

*Solution:*

Since the successive terms of the series have a constant difference, the given series is an arithmetic series.

Formula for sum of an arithmetic series =

Where,

n = number of terms in the series,

a = first term of the series,

d = common difference of the series.

d = a_{2} – a_{1}

= 6 – 3

d = 3

Finding n, we know that the nth term is 42

Therefore, 42 = a + (n-1) d

= 3 + (n-1) * 3

39 = (n-1) * 3

n = 1 + 13 = 14.

Therefore, the answer is 315.

The above approaches help to solve a problem of arithmetic progression.

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