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 hereDefinition 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 APThree 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 followsS = 1 + 2 + 3 + ................... + 99 + 100And then, reversed the numbers to writeS = 100 + 99 + ................. + 2 + 1Then he add both the sequences2S = (100 + 1) + (99 + 2) + (98 + 3) + ............... + (2 + 99) + (1 + 100)2S = 101 + 101 + 101 + .................... + 101 + 101 to 100 times2S = 101 X 100    ⇒  $S=\frac{100\times 101}{2}=5050$If we take 100 = n, then n+1 = 101,So we can derive the formula for the sum of first n natural number$S_{n}=\frac{n(n +1)}{2}$With the help of above explanation we can derive the formula for finding the sum of n terms of an APS = a + (a + d) + (a + 2d)+...................... + a + (n - 1)dS = a + (n - 1)d + [a + (n - 2)d] +...............+ (a + d) + aAdding these two we get2S = [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 times2S = n X [2a+(n-1)d]$S=\frac{n}{2}\left [ 2a+(n-1)d \right ]$$S=\frac{n}{2}\left [ a+\left \{ a+(n-1)d \right \} \right ]$$S=\frac{n}{2}\left [ a+l \right ]$ Sum of the first n terms of AP is $S_{n}=\frac{n}{2}\left [ 2a+(n-1)d \right ]$ 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  $S_{n}=\frac{n}{2}\left [ a+l\right ]$ 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$S_{n}=\frac{n}{2}\left [ 2l-(n-1)d\right ]$nth term of an AP sequence is given by $a_{n}=S_{n}-S_{n-1}$Sum of n even natural numbers is given by: $n(n+1)$Sum of first n odd natural numbers is given by  $= n^{2}$

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.

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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  $b=\frac{a+c}{2}$
Explanation:-

b - a = d and c - b = d

⇒ b - a = c - b

⇒2b = a + b

$\Rightarrow b = \frac{a+c}{2}$

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.

THANKS FOR YOU VISIT