Class 9 – Progressions
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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 = <an> 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

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+1 – an is independent of n, the given sequence is a A.P. otherwise it is not an A.P.
Example: Show that the sequence <a_{n}> defined by a_{n} = 4n + 5 is an A.P. Also, find it
common difference.
Solution:We have a_{n} = 4n + 5. Replacing n by (n + 1) weget
Now,
Clearly,an+1 – an 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+1 – anis 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 nonzero 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 nonzero 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.