ARITHMETIC PROGRESSIONARITHMETIC PROGRESSION

MADE BY :-MADE BY :-NAME :- PRANAV GHILDIYALNAME :- PRANAV GHILDIYAL

CLASS :- X DCLASS :- X D

What is Arithmetic ProgressionsWhat is Arithmetic Progressions• An arithmetic progression is a list of numbers in which each term is obtained

by adding a fixed number to the preceding term except the first term.

• This fixed number is called the common difference of the AP. Remember that it can be positive, negative or zero.

• Each of the number in the A.P. is known as term.• A, a+d, a+2d, a+3d with a as first term and common difference d is known as general form of an A.P.

• There are two types of A.P. FINITE- which has only finite number of terms. INFINITE- which has infinite number of terms.

Pattern OR A.P.• A pattern is a type of theme of recurring events or objects, sometimes

referred to as elements of a set of objects. It is different from an A.P. as A.P. has a common difference between two terms but pattern doesn’t have.

Types of progressionsArithmetic Progression

• an arithmetic progression (AP) is an arithmetic progression (AP) is a sequence of numbers such that the a sequence of numbers such that the difference between the consecutive difference between the consecutive terms is same. For instance, the terms is same. For instance, the sequence 3, 5, 7, 9, 11, 13, … is an sequence 3, 5, 7, 9, 11, 13, … is an arithmetic progression with common arithmetic progression with common difference 2.difference 2.

• If the initial term of an arithmetic If the initial term of an arithmetic progression is aprogression is a11 and the common and the common difference of successive members is d, difference of successive members is d, then the nth term of the sequence is then the nth term of the sequence is given by: given by: aann = a +(n-1)d = a +(n-1)d

• There are two types of A.P. FINITE- There are two types of A.P. FINITE- which has only finite number of terms. which has only finite number of terms. INFINITE- which has infinite number of INFINITE- which has infinite number of terms.terms.

Geometric progressionGeometric progression• a geometric progression is a geometric progression is a sequence of numbers where each a sequence of numbers where each term after the first is found by term after the first is found by multiplying the previous one by a multiplying the previous one by a fixed non-zero number called fixed non-zero number called the common ratio. For satisfying the common ratio. For satisfying G.P., three consecutive terms a ,b G.P., three consecutive terms a ,b

and and c shall satisfy the following c shall satisfy the following equation: bequation: b22 = ac For example, the = ac For example, the sequence 2, 6, 18, 54, ... is a sequence 2, 6, 18, 54, ... is a geometric progression with common geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common a geometric sequence with common ratio 1/2. The sum of the terms of a ratio 1/2. The sum of the terms of a geometric progression is known as geometric progression is known as a geometric series.a geometric series.

FINITE AND INFINITE A.P.FINITE AND INFINITE A.P.Finite

• Arithmetic progressions which have fixed number of terms is defined as finite arithmetic progression.

• Example:-• 1, 3, 5, 7, c.d. = 2• –7, –3, 1, 5, 9 c.d. = 4• 8, 5, 2, –1, –4 c.d. = –3

• Last term is known

Infinite

• Arithmetic progressions which have unfixed number of terms is defined as infinite arithmetic progression.

• Example:-• 1,8,15,22………….. c.d = 7• 11,22,33…………… c.d. = 11

• Last term is unknown

ARITHMETIC MEANARITHMETIC MEAN• When three numbers are in A.P., the middle term is called arithmetic mean (AM) of

the two others.• Let a and b are two numbers.• And A be the arithmetic mean between two numbers.• So a A b are in A.P.• or, A – a = b – A• or, 2A = a + b• A =a+b/2• So, in general

• A.M. = Sum of the numbers/2

FORMULAS IN THE CHAPTERFORMULAS IN THE CHAPTER

• aan = a +(n-1)dn = a +(n-1)d

• SSn =n/2[2a+(n-1)d]n =n/2[2a+(n-1)d]

OROR

• SSnn= (a+a= (a+ann))The sum of first n positive integer is The sum of first n positive integer is

given bygiven by• ssnn=n(n+1)/2=n(n+1)/2

Derivation of aDerivation of annthe second term a2 = a + d = a + (2 – 1) dthe third term a3 = a2 + d = (a + d) + d = a + 2d = a + (3 – 1) dthe fourth term a4 = a3 + d = (a + 2d) + d = a + 3d = a + (4 – 1) d

we can say that the nth term an = a + (n – 1) d.So, the nth term an of the AP with first term a and common difference d is

given by an = a + (n – 1) d.

Derivation of SDerivation of SnnS = a + (a + d ) + (a + 2d) + . . . + [a + (n – 1) d ] ............................................... (1)

Rewriting the terms in reverse order, we have

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

On adding (1) and (2), term-wise. we get

2S =[2a +(n-1)d] + [2a +(n-1)d] + …….[2a +(n-1)d] + [2a +(n-1)d [n times]or, 2S = n [2a + (n – 1) d ] (Since, there are n terms)

or, S = n/2 [2a + (n – 1) d ]

So, the sum of the first n terms of an AP is given by

S = n/2 [2a + (n – 1) d ]

i.e., S =n/2 (a + an ) ………………………………………………………………… (3)

Now, if there are only n terms in an AP, then an = l, the last term. From (3), we see that

S =n/2 (a + l )

Finding aFinding ann,, S Snn, n, d, a, n, d, a

GIVEN THATGIVEN THAT

an = a + (n − 1)d ∴ 50 = 5 + (n − 1)3

45 = (n − 1)3 15 = n − 1 n = 16

GIVEN THATGIVEN THAT an = a + (n − 1) d, ∴ a13 = a + (13 − 1) d

35 = 7 + 12 d 35 − 7 = 12d 28 = 12d

GIVEN THATGIVEN THAT

find n and an.

90 = n [2 + (n − 1)4]90 = n [2 + 4n − 4]90 = n (4n − 2) = 4n2 − 2n4n2 − 2n − 90 = 04n2 − 20n + 18n − 90 = 04n (n − 5) + 18 (n − 5) = 0

(n − 5) (4n + 18) = 0Either n − 5 = 0 or 4n + 18 = 0n = 5 or However, n can neither be negative nor fractional.Therefore, n = 5

a = 3 n = 8

S = 192

a = 3 n = 8

S = 192as 192 = 4 [6 + 7d]48 = 6 + 7d42 = 7dd = 6

Find d

GIVEN THAT

GIVEN THAT

(16) × (2) = a + 28

32 = a + 28

a = 4

GIVEN THAT GIVEN THAT

Check whether 301 is a term of the list of numbers 5,11,17,23….the given list of numbers is an AP. With a = 5 and d = 6.the given list of numbers is an AP. With a = 5 and d = 6.

Let 301 be a term, say, the Let 301 be a term, say, the nth term of the this AP.nth term of the this AP.We know that We know that aann = a + (n – 1) d = a + (n – 1) dSo, 301 = 5 + (So, 301 = 5 + (n – 1) × 6n – 1) × 6i.e., 301 = 6i.e., 301 = 6n – 1n – 1So, So, n = 302/6 OR 151/3n = 302/6 OR 151/3But But n should be a positive integer . So, 301 is not a n should be a positive integer . So, 301 is not a

term of the given list of term of the given list of numbers.numbers.

EXAMPLES OF A.P. IN REAL LIFEEXAMPLES OF A.P. IN REAL LIFE• Increament in salary of a preson in rupees not in

percentage• If we take installment as increasing order or

decreasing order of rupees this also is in AP• Days in a month..... 1,2,3,4,......

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