Simple approach to learn Sequence and series

Sequence and Series

Sequence

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

a1, a2, a3, . . ., an, . . .

Series

If a1, a2, a3, . . ., an, . . .is a sequence,

the expression a1+a2+a3+ . . . +an+ . . .

is called a series.

Arithmetic Progression

A sequence is called an arithmetic progression (A.P.) if the difference between any term and the previous term is constant.

The constant difference, generally denoted by d is called the common difference.

a1 = a,  a2 = a+d, a3 = a+d+d  = a+2d ,

a4 = a+d+d+d  = a+3d 

Choose Right term!!

Geometric Progression

A sequence is called a geometric

progression (G.P.) if the ratio between any

term and the previous term is constant.

The constant ratio, generally denoted by r is

called the common ratio.

a1 = a,  a2 = ar, a3 = ar2,  a4 = ar3

Choose Right term!!

Harmonic Progression

Arithmetic Mean

A is the A.M. of two numbers a and b

Û a, A, and b are in A.P.

\ A-a = b-A

\ 2A = a+b

Inserting n A.M between 2 numbers

Geometric Mean

G is the G.M. of a and b

Û G2 = ab

Inserting n G.M between 2 numbers

Harmonic Mean

Inserting n H.M between 2 numbers

Relation Between A.M., G.M. And H.M.

Let A, G and H be the arithmetic, geometric and harmonic means of two positive numbers a and b.

G2=AH

Arithmetico-Geometric Progression

A sequence is called an arithmetico-geometric progression (A.G.P.) if the nth term is a product of the nth term of an A.P. and the nth term of a G.P.

Sigma notation

Logarithmic Series

Exponential Series

Rank of a Word in Permutations

Rank of a word

# Word having distinct alphabets 

In this we are going to find rank of a word with distinct alphabets.

In permutations we often come across questions on number of words formed using letters of the given word.Here we are calculating rank of a word i.e. the place of a specific word when words are arranged as arranged in dictionary.

Consider a word SUMIT in which there are 5 alphabets and all are distinct.
From letters of the word SUMIT , there would be 5! = 120 different meaningful and meaning less words can be formed.

If these words are arranged alphabetically i.e. as arranged in dictionary then the word SUMIT comes at certain place i.e.rank of a word SUMIT.

Procedure:

3              5              2              1              4              

S              U             M            I               T                   

Then starting from left i.e from letter S, count the numbers  to the right side which are less than 3 i.e.number corresponds to letter S which is third in alphabetical order of the word SUMIT.

So there are two numbers less than 3 i.e. 1 and 2 so write 2 below S as shown.

3              5              2              1              4             

S              U             M            I               T                   

2

In the same way for next alphabet U the corresponding number is 5. So there are three numbers less than 5 to the right of U i.e.  2,1 & 4.So write 3 below U  and so on for other alphabets.

3              5              2              1              4             

S              U             M            I               T                   

2              3              1              0              0 

Now from rightmost end write factorials start from 0! in increasing order up to last alphabet. 

3              5              2              1              4              

S              U             M            I               T                   

2              3              1              0              0

4!            3!             2!            1!              0!

Now multiply corresponding factorial with number written above it to get some results.

3              5              2              1              4             

S              U             M            I               T                   

2              3              1             0               0

*              *              *              *              *

4!            3!            2!            1!              0!

The multiplication will give following results 

3              5              2              1              4              

S              U             M            I               T                   

2              3              1              0              0

*              *              *              *              *

4!            3!            2!             1!             0!

48           18           2               0              0

Now add numbers 48 + 18 + 2 + 0 + 0 = 68 .Add one to this addition to get a total of 69.

Rank of word SUMIT is 69. 

SPECIAL TYPES OF NUMBERS

1] PALINDROME NUMBERS

A palindromic number is a number that is the same when written forwards or backwards, i.e., of the form . 

For example numbers like 121,4004,66..etc

2]  ARMSTRONG NUMBERS

An Armstrong number of three digits is an integer such that the sum of the cubes of its digits is equal to the number itself. 

For example, 371 is an Armstrong number since 3³ + 7³ + 1³ = 371.

3] AUTOMORPHIC NUMBERS

In mathematics an automorphic number is a number whose square "ends" in the same digits as the number itself.

For example, 5² = 25, 6² = 36, 26² = 676,  35² = 1225...etc


4] NEON NUMBERS

A number is said to be a Neon Number if the sum of digits of the square of the number is equal to the number itself. 

For Example 9 is a Neon Number. 9*9=81 and 8+1=9.Hence it is a Neon Number.

5] AMICABLE NUMBERS

Amicable numbers are a pair of numbers with the following property: the sum of all of the proper divisors of the first number (not including itself) exactly equals the second number while the sum of all of the proper divisors of the second number (not including itself) likewise equals the first number.

For example let's show that 220 & 284 are amicable numbers:
First we find the proper divisors of 220:
1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110
If you add up all of these numbers you will see that they sum to 284.
Now find the proper divisors of 284:
1, 2, 4, 71, 142
These sum to 220, and therefore 220 & 284 are amicable numbers.

Discoveries in Mathematics

1.Sir Isaac Newton Calculus 

Isaac Newton discovered calculus in the mid-17th century.











2. James Sylvester -- Matrix Theory

James Sylvester discovered Matrix theory in 1850.

3. Josiah Willard Gibbs -- Vectors 

Josiah  Williard Gibbs discovered vector theory in mathematics
 in late 19th centuary.

4. Blaise Pascal -- Theory of Probability

Blaise Pascal discovered theory of Probability
 in 1654.

5. John Napier -- Logarithms

John Napier discovered Logarithms
 in 1614.

6.  Seki Takakazu -- Determinants

Seki Takakazu discovered Determinants
 in 1683.

7.  Gerolamo Cardano -- Complex number

Gerolamo Cardano  discovered complex number
 in 1545.