Sequence and Series Formula

Sequence and Series

We can define a sequence as an arrangement of numbers in some definite order according to some rule.

We can commonly represent sequence as,

x1,x2,x3,......xn. where 1,2,3 are the position of the numbers and n is the nth term 

Whereas, series is defined as the sum of sequences.

Example: 1+2+3+4+.....+n, where n is the nth term 

Series and sequence are the concepts that are often confused.

Suppose we have to find the sum of the arithmetic series 1,2,3,4 ...100. We have to just put the values in the formula for the series. Let us memorize the sequence and series formulas.


Types of Sequence

There are three types of sequence

  • Arithmetic Sequences

  • Geometric Sequence

  • Fibonacci Sequence


Arithmetic Sequence

Any sequence in which the difference between every successive term is constant then it is called Arithmetic Sequences. 

Example

3,  6,  9,  12, 15,  18, 21……..

  +3 +3  +3  +3  +3  +3

Here the difference between the two successive terms is 3 so it is called the difference.

The difference is represented by “d”.

In the above example, we can see that a1 =0 and a2 = 3.

The difference between the two successive terms is

a2 – a1 = 3

a3 – a2 = 3

In an arithmetic sequence, if the first term is a1 and the common difference is d, then the nth term of the sequence is given by:

an= a1+ (n−1) d

 

Geometric Sequences

A sequence in which every successive term has a constant ratio between them then it is called Geometric Sequence. 

Example

1, 4, 16, 64…...

Here

a1 =1

a2 = 4 = a1(4)

a3 = 16 = a2(4)

Here we are multiplying it with 4 every time to get the next term. Here the ratio is 4 .

The ratio is denoted by “r”.

an = an-1⋅r or 

an = a1⋅rn−1

an = an-1x r 


Fibonacci Sequence

By adding the value of the two terms before the required term, we will get the next term. Such type of sequence is called the Fibonacci sequence. There is no visible pattern.

Example

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 54, 88, 142  ...

In the above sequence, we can see

a1 =0 and a2 = 1

a3 = a2 + a1 = 0 + 1 =1

a4 = a3 + a2  = 1 + 1 =2 and so on.

So the Fibonacci Sequence formula is

an = an-2 + an-1, n > 2

This is also called the Recursive Formula.

 

Series

The summation of all the numbers of the sequence is called Series. Generally, it is written as Sn.

Example

If we have a sequence 1, 4, 7, 10, …

Then the series of this sequence is 1 + 4 + 7 + 10 +…

The Greek symbol sigma “Σ” is used for the series which means “sum up”.

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The series 4 + 8 + 12 + 16 + 20 + 24 can be expressed as  \[\sum_{n=1}^{6}4n\]. We read this expression as the sum of 4n as n ranges from 1 to 6. 


Types of Series

There are different types of series-

Arithmetic Series

An arithmetic series is the sum of a sequence ai, i = 1, 2,....n which each term is computed from the previous one by adding or subtracting a constant d. Therefore, for i>1 

ai = ai-1 + d = ai-2 + d=............... =a1 + d(i-1)

where a is the first term and d is the difference between the terms which is known as the common difference of the given series.

The Formula of Arithmetic Series

The formula for the nth term is given by if a is the first term, d is the difference and n is the total number of the terms, then the 

an = a + (n - 1) d

Sum of an Arithmetic Series

Sn = n/2 [2a + (n-1)d]

Geometric Series

Geometric series is the sum of all the terms of the geometric sequences i.e. if the ratio between every term to its preceding term is always constant then it is said to be a geometric series.

The Formula of Geometric Series

In general, we can define geometric series as

\[\sum_{n=1}^{∞}ar^{n}\] = a + ar + ar2 + ar3 + …….+ arn

Where a is the first term and r is the common ratio for the geometric series.

an = a1 r n - 1 

Then the formula for the nth term is

Sum of Geometric Series

Sn  = \[\frac{a(1-rn)}{1-r}\]


Geometric Progression (G.P.)

The sequence of numbers in which the next term of the sequence is obtained by multiplying or dividing the preceding number with the constant number is called a geometric progression. The constant number is called the common ratio. It is also known as Geometric Sequences.

a, ar, ar2, ar3, …, arn


Arithmetic Mean

The arithmetic mean is the average of two numbers. If we have two numbers n and m, then we can include a number A  in between these numbers so that the three numbers will form an arithmetic sequence like n, A, m.

In that case, the number A is the arithmetic mean of the numbers n and m.

A = (n + m) / 2


Geometric Mean

Geometric Mean is the average of two numbers. If p and q are the two numbers then the geometric mean will be 

GM =\[\sqrt{pq}\]


Harmonic Mean

By the harmonic mean definition, harmonic mean is the reciprocal of the arithmetic mean, the formula to define the harmonic mean “H” is given as follows:

Harmonic Mean(H) = n / [(1/x1)+(1/x2)+(1/x3)+…+(1/xn)]

Where,

n is the total number of terms

 x1, x2, x3,…, xn are the individual values up to nth terms.


Solved Example

Example 1: What will be the 6th number of the sequence if the 5th term is 12 and the 7th term is 24?

Solution: As the two numbers are given so the 6th number will be the Arithmetic mean of the two given numbers.

AM = 12 + 24 / 2

= 36/2

= 18

Hence the 6th term will be 18.

Example 2: Find the geometric mean of 2 and 18.

Solution: Formula to calculate the geometric mean.

p = 2 and q = 18

GM = \[\sqrt{pq}\]

= \[\sqrt{2 × 18}\]

=  \[\sqrt{36}\]

= 6


Quiz Time

What is the ninth term of the geometric sequence 3, 6, 12, 24, ...?

What is the sum of the first ten terms of the geometric sequence 5, 15, 45, ...?

FAQ (Frequently Asked Questions)

1. Difference Between Sequence and Series

Ans. There is a lot of confusion between sequence and series, but you can easily differentiate between Sequence and series as follows:

  • A sequence is a particular format of elements in some definite order, whereas series is the sum of the elements of the sequence.

  • In sequence order of the elements are definite, but in series, the order of elements is not fixed.

  • A sequence is represented as 1,2,3,4,....n, whereas the series is represented as 1+2+3+4+.....n.

  • In sequence, the order of elements has to be maintained, whereas in series the order of elements is not important.

2. What is an Arithmetic Series?

Answer: An arithmetic series is what you get when you add up all the terms of a sequence. The resulting values are called the "sum" or the "summation". Series is indicated by either the Latin capital letter "S'' or else the Greek letter corresponding to the capital "S'', which is called "sigma" (SIGG-muh): written as Σ.

To show the summation of tenth terms of a sequence {an}, we would write as,

Σ10n=1an

Where "n = 1" is called the "lower index", it represents that the series starts from 1 and the “upper limit” is 10 it means the last term will be 10. And "an" stands for the terms that we'll be adding. It is read as "the sum, from n equals one to ten, of a-sub-n".