# Class 9 – Progressions

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Progressions
• Arithmetic progression
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Progressions
• Sum to n terms of an A.P.
• Arithmetic mean
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## Study Material

 Introduction: A sequence is a function whose domain is the set N of natural numbers. It is customary to denote a sequence by a letter ‘a’ and the image of under ‘a’ by an. The images  are known as first term, second term, …nth term, ….respectively of the sequence. If  is the nth term of a sequence, ‘a’ then we write a = Representation of a sequence: There are several ways of representing a sequence. One way to represent a sequence is to list its first few terms till the rule for writing down other terms becomes clear. For example, 1, 3, 5. . . .  is a sequence whose nth term is (). Another way to represent a real sequence is to give a rule, the sequence 1, 3, 5, 7, ….. can be written as Sometimes we represent a sequence by using a recursive relation. For example, the Fibonacci sequence is given by The above recursive relation gives the terms of this sequence as relation gives the terms of this sequence as 1, 1, 2, 3, 5, 8,…………..   Series: If is a sequence then the expression  is a series.   Example: If  sequence then  is a series A series is finite or infinite according as the number of terms in the sequence is finite or infinite.   Progression: A progression is a sequence whose terms follow a certain pattern i.e., the terms are arranged under a definite rule of addition and subtraction.

Arithmetic Progression(A.P.): A sequence is called an arithmetic progression, if the difference of a term and the previous term is always same.

.

The constant difference, generally denoted by d is called the common difference.

Example: 1, 4, 7, 10,……is an A.P whose first term is 1 and the common difference
= 4 – 1 = 3.

Example:2, 7, 12, 17, 22,…….. is an A.P whose first term is 2 and the common difference  = 7 – 2  = 5.

Example: -3, -7,-11, -15,….. is an A.P whose first term is -3 and the common difference
is – 7 – (-3) = -7 + 3 = -4.

Algorithm to determine whether a sequence is an A.P. or not when it nth term is given.

Step 1: Obtain an(then nth term of the sequence)

Step 2: Replace n by n + 1 is an to get an+1.

Step 3: Calculate an+1- an

Ifan+1an is independent of n, the given sequence is a A.P. otherwise it is not an A.P.

Example: Show that the sequence <an> defined by an = 4n + 5 is an A.P. Also, find it
common difference.

Solution:We have an = 4n + 5. Replacing n by (n + 1) weget

Now,

Clearly,an+1an is independent of n and is equal to 4.

So,the given sequence is a A.P. with common difference 4.

Example: Show that the sequence <an > defined by  is not an A.P.

Solution: We have  replacingn by (n+1) in a. we obtain

Now,

Clearly, an+1anis not independent of n and is therefore not constant. So, the given sequenceis not A.P.

The sequence<an > is an A.P. If its nth term is an linear expression in n and in such a case the common difference is equal to the coefficient of n.

Proof: Let <an > be asequence such that it nth term is a linear expression in n i.e.,

where,A,B are constants.

Clearly, an+1- an is independent of n and is therefore a constant. So, the sequence <an >  is an A.P. with common difference A.

General term of A.P.: Leta be the first term and d be the common difference of an A.P

The second term

Fromthe above pattern, we can observe that

Example: If the first term of an A.P. is -3 andthe common difference is ,find its 8th term and the nth term.

Solution: Given

nth term of an A.P. from the end: Let ‘a’ be the first term and d be the common difference of an A.P having m terms. The  term from the end is  term from the beginning.

So,term from the end =

Important Result: In an A.P the sum of the terms equidistant from the beginning and end is always same and equal to the sum of first and last terms.

Solution: Let bean A.P with common difference ‘d’ then,  term from the beginning  and

(kthterm from beginning )+(kth term from the end )

Thusforall  Hence,sum of terms equidistant from beginning and end is always same and equal to the sum of first and last terms.

Properties of Arithmetic Progression:

1.If a same non-zero constant is added or subtracted from each term of an A.Pthen the
resulting sequence is also an A.P with the same common difference.

2. If each term of a given A.P ismultiplied or divided by a non-zero constant k, then the
resulting sequences is also an A.P. with common difference ,where d is the
common difference of the given A.P.

3.Three numbers a, b, c are in A.P.  if and only if 2b = a + c.

Proof: Let a,b, c be in A.P. Then,

b- a = common difference and c – b = common difference

_ b – a = c – b   _ 2b = a + c

Conversely, let a, b, c are three numbers such that 2b = a + c. Then we have to showthat
a, b, c are in A.P.

We have 2b = a + c  _b – a = c – b  _ a, b, c are in A.P

Example:If  inA.P. find the value of k.

Solution: arein A.P.

Example:Find the 10th term of the A.P.: 2, 4, 6,….

Solution:Here the first term (a) = 2 and common difference d = 4 – 2 = 2

Usingthe formula , wehave

Hence,the 10th term of the given A.P. is 20.

Example: The 10th term of an A.P. is -15 and 31st term is -57, find the 15thterm.

Solution:Let a be the first term and d be the common difference of the A.P. Then from the formula.

Wehave,

Solve equation (1) and (2) to get the values of a and d.

Subtracting(1) from (2), we have

Againfrom (1),

Now

Example: Which term of the given A.P.: 5, 11, 17,… is 119?

Solution:Here a = 5, d = 11 – 5 = 6

Weknow that

Therefore,119 is the 20th term of the given A.P.

Example: Is 600 a term of theA.P. 2, 9, 16….?

Solution:Here, a = 2, and d = 9 – 2 = 7

Let 600 be the nth term of the A.P. We have

Since n is a fraction, it cannot be a term of the given A.P. Hence, 600 is not a term of the given A.P.

Example: The common difference of a A.P is 3 and the 15th term is 37. Find the first
term.

Solution:Here, d = 3,

Letthe first term be a. We have

Thus,first term of the given A.P. is -5.

Example: If p times the pthterm of an AP is equal to q times the qth term, prove that its
(p+q)th term is zero, provided

Solution: We have

Since,LHS is nothing but  term,therefore,

Example:In the arithmetic progression 2, 5, 8,….. upto 50 terms and 3, 5, 7, 9,… upto60 terms, find how many terms are identical ?

Solution: Let mthterm of the first A.P be equal to nth term of second A.Pthen,

Corresponding to each value of k,we get a pair of identical terms. Hence there are 20 identical terms in the twoA.P.’s.