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CBSE Class 10 Maths Formulas Chapter-05 | Arithmetic Progression
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ARITHMETIC PROGRESSION
Class 10 Chapter 5
Class 11 Chapter 9
Sequence and Series
|
Sequence :
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. |
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, ...............
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, ...............
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 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 + 5dNOTE:- 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

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.
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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