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(t_n) = 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.

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ARITHMETIC MEAN 

If between two quantities a and b we have to insert n quantities A₁, A₂, A₃, A₄, ………, A_n be such that A, A₁, A₂, A₃, A₄, ………, A_n, b form an AP then we say that A₁, A₂, A₃, A₄, ………, A_n 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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