Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

Arithmetic Progression | Chapter 5 Class 10

ARITHMETIC PROGRESSION
Class 10 Chapter 5
Class 11 Chapter 9


Sequence and Series

 Main Points to be discussed here

  • Definition of Sequence, Series and Progression.
  • Concept of Arithmetic Progression, General AP Sequence, General Term, Sum of n terms in an AP and Arithmetic Mean.
  • Finite and Infinite AP
  • Three terms in AP, Four terms in AP and five terms in an AP.
  • Properties of Arithmetic Mean

Sequence :
An arrangement of numbers in a definite order according to some rule is called a sequence. The terms of the sequence are denoted by t1, t2, t3, t4, ……….. or a1, a2, a3, a4, ………………….
Example: 2, 4, 6, 10, 12, 14, ................

Series :

If the terms of a sequence are connected by plus (or minus) sign, is called a series.
Example: 2 + 4 + 6 + 10 + 12 + 14 + ................

Progression:

A sequence following some definite rule is called a progression.

Arithmetic Progression

A sequence is called arithmetic progression if the difference of a term and its previous term is always same.

A sequence a1, a2, a3 , a4,  a5 ………, an , an+1 ……… is called arithmetic progression if 

an+1 = an + d, 

where a1 is called first term and d is called the common difference.


For example: Let us suppose a sequence :
3, 6, 9, 12, 15, ............
6 - 3 = 3, 9 - 6 = 3, 12 - 9 = 3, ..........
t1

Here we find that difference of any two consecutive terms is always remain same (3).
When difference of any two consecutive terms of a sequence remain same throughout the sequence, then that sequence is said to be in AP.

General Arithmetic Progression:-

a , a + d, a + 2d, a + 3d, a + 4d +.............+ a + (n - 1)d

Where

First Term(
t1) = a

Second Term(
t2) = a + d

Third Term(
t3) = a + 2d
..............................
.................................
Last term(
tn) = a + (n-1)d

nth term of sequence is   tn = a + (n-1)d

Where common difference “d” is given by:  d = an – an-1

nth terms of an AP from the end of the sequence is:   l – (n-1)d,  

where l is the last term of the sequence.


Finite Arithmetic Progression : When number of terms of an AP sequence are countable then AP is called Finite AP.
For example : 5, 10, 15, ............ 65.
Infinite Arithmetic Progression: When number of terms of an AP sequence are uncountable then AP is called infinite AP.
For example : 5, 10, 15, ...............

Sum of First n terms of an AP  

Here is a very interesting story related to the origin of this topic.


Gauss was a great mathematician. When he was just 10 years old, teacher tell all the students of his class to add all the numbers from 1 to 100. He immediately replied that the answer is 5050. Can you guess, how did he do?

We are here explain the method used by Gauss at that time :

He wrote the numbers as follows

S = 1 + 2 + 3 + ................... + 99 + 100

And then, reversed the numbers to write

S = 100 + 99 + ................. + 2 + 1

Then he add both the sequences

2S = (100 + 1) + (99 + 2) + (98 + 3) + ............... + (2 + 99) + (1 + 100)

2S = 101 + 101 + 101 + .................... + 101 + 101 to 100 times

2S = 101 X 100    ⇒  
If we take 100 = n, then n+1 = 101,

So we can derive the formula for the sum of first n natural number

With the help of above explanation we can derive the formula for finding the sum of n terms of an AP

S = a + (a + d) + (a + 2d)+...................... + a + (n - 1)d

S = a + (n - 1)d + [a + (n - 2)d] +...............+ (a + d) + a

Adding these two we get

2S = [a + a + (n - 1)d] + [(a + d) + a + (n - 2)d] + ................ + [a + (n - 1)d + a]

2S = [2a + (n - 1)d] + [2a + (n - 1)d] + ........................... n times

2S = n X [2a+(n-1)d]





 
Sum of the first n terms of AP is  

Where "a" is the first term and "d" is the common difference of the given AP sequence.

Sum of the first n terms of an AP is   
Where "a" is the first term and "l" is the last term of the given AP sequence.
Sum of the first n terms from the end of the AP sequence is given by


nth term of an AP sequence is given by



Sum of n even natural numbers is given by: 

Sum of first n odd natural numbers is given by  



Three terms in AP can be taken as :- a - d, a, a + d

Four terms in AP can be taken as :- a - 3d, a - d, a + d, a + 3d

Five terms in AP can be taken as :- a - 2d , a - d , a , a + d , a + 2d

Six terms in AP can be taken as :- a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d

NOTE:- Next topic is for the students after 10th standard in CBSE Board.

********************************************************  


ARITHMETIC MEAN 
If between two quantities a and b we have to insert n quantities A1, A2, A3, A4, ………, An be such that A, A1, A2, A3, A4, ………, An, b form an AP then we say that A1, A2, A3, A4, ………, An are n arithmetic means between a and b
If a, b, c are in AP then middle term(b) is called the arithmetic mean and is given by  
Explanation:-

b - a = d and c - b = d

⇒ b - a = c - b

⇒2b = a + b 


PROPERTIES OF A.P.

1) If a constant is added to or subtracted from each term of an A.P. , then the resulting sequence is also an A.P. , with the same common difference.

2) If each term of an A.P. is multiplied or divided by an non-zero constant k, then the resulting sequence is also an A.P. with common difference kd or d/k, where d is the common difference of the given A.P.

3)In a finite A.P. the sum of the terms equidistant from the beginning and end is always same and is equal to the sum of first and last term.

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