Class 10 – progressions
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Introduction:A collection of numbers arranged in a definite order according to some definite rule or pattern is called a sequence.
i) is a sequence which is having a pattern of squares of natural numbers.
ii) 2, 4, 6, 8, ….. is a sequence whose definite rule is even numbers.
Arithmetic progression:A sequence is called an arithmetic progression if every term is obtained by adding or subtracting a fixed number to its preceeding term except the first term.
The fixed number is called the common difference of arithmetic progression.
Thus, if a is the first term of A.P and d is the common difference, then the terms of A.P are . It is called as general form of A.P.
Here is second term, is third term and so on.,
Example: Identify which of the following are in A.P.
Solution:i) The sequence is
Here first term
The common difference is same in each case, so given sequence is in A.P.
ii) In sequence
Since common difference is not constant the given sequence is not in A.P.
iii) In the sequence
The common difference is not same
The given sequence is not in A.P.
nth term of an A.P:
Let be an AP whose first term is and the common difference is d.
Then, the second term
The third term
Looking at the pattern, we can say that the nth term
So, the nth term an of the AP with the first term a and common difference d is given by
is also called the general term (or) last term of the A.P
If there are m terms in the AP then represents the last term which is sometimes also denoted by l. Using the above formula, we can find different terms of an arithmetic progression.
(i) If a constant term is added or subtracted to an each term of an A.P, then the resulting sequence is also an A.P.
(ii) If each term of an A.P. is multiplied (or) divided by a non-zero constant. Then the resultant is also in A.P.
Important facts related to AP:
1. If a fixed constant c is added to (or subtracted from) each term of a given A.P., then the resulting sequence is also an A.P. with the same common difference as that of the given A.P.
2. If each term of an A.P. is multiplied by (or divided by) a non-zero fixed constant c, then the resulting sequence is also an A.P. with common difference c times (or) 1/c times the previous A.P.
3. If …. And …… are two AP’s, then …….is also an AP with common difference .
4. A sequence obtained by multiply or dividing of corresponding terms of two AP is not an A.P.
5. Three numbers in AP can be taken as a – d, a, a + d, five numbers in AP can be taken as a – 2d, a – d, a, a + d, a + 2d, where the common difference is d.
6. Four numbers in AP can be taken as a – 3d, a – d, a + d, a + 3d, where common difference is 2d.
7. A sequence is an AP, if its term is of the form An + B, i.e., a linear expression in n in such a case the common difference is A i.e., the coefficient of n.
8. The common difference of AP can be zero, positive or negative.
Arithmetic mean (A.M): If a, b, c are in A.P. the ‘b’ is called the arithmetic mean (A.M) between a and c.
Example: The AM of 2 and 6 is 4
If there are ‘n’ AMs between ‘a’ and ‘b’ then common difference (d) =
Example:If 3 AMs are inserted between 5 and 21, then find the AMs.
Solution:Let 3 AMs are
∴ Three AMs are 9, 13 and 17.
Aliter: a = 5, b = 21
Common difference (d) =
We need 3 AM’s, therefore n = 3
AM’s are 5 + 4, 9 + 4, 13 + 4
∴ Three AM’s are 9, 13 and 17.
Example:Find the 10th term of the AP: 9, 6, 3 ….
∴ 10th term of the given AP is -18.
Example:Which term of the AP: 5, 8, 11…….is 155
Example: Check whether 723 is a term of the A.P 7,13,19,….
a = 7; d = 12 – 7 = 6
Let us assume
Here n is not a positive integer
∴ 723 is not a term of the given A.P.
Example: How many three digit numbers are divisible by 4.
Solution: Three digit numbers divisible by 4 are 100, 104, 108, ………996
The obtained series is in A.P
Where a = 100 and d = 4
and we have last term
996 = a + (n-1)d = 100 + (n-1) 4
896 = (n-1) 4
n = 224+1 = 225
∴ There are 225 three digit numbers divisible by 4.
Example:Find the 10th term from the last of the AP series given below
8, 5, 2……….,-64
a = 8, d = 5 – 8 = -3, l = -60
To find the 10th term from the last term, we will find the total number of terms in the AP
-64 = 8 + (n-1) -3
-72 = (n-1) -3
(n -1) = 24 ⇒ n =25
So, there are 25 terms in the given AP
The 10th term from the last will be the 16th term of the series.
∴ 10th term from the end is -37.
Example:Find the first three terms of the sequence whose general term is
Example: Write the first four terms of the sequence given by
Putting n = 3 in (1) , we get
Putting n = 4 in (1), we get
Example: What is the nth term of the sequence 3, 6, 9 ….?
Solution: Here a = 3, d = 3 (given series form an A.P.)
Example:m arithmetic means have been inserted between 1 and 31 in such a way that the ratio of the 7th and the (m – 1)th means is 5 : 9, find the value of m.
Solution:Let be the m arithmetic means between 1 and 31
Then, 1, , 31 are in A.P.
Let d be the common difference of this A.P.
From eqs. (i) and (ii), we get
Example: If in an A.P., the pth term is q and the term is 0, then find the qthterm.
Arithmetic progression is abbreviated as AP
If each term of an A.P. be : (i) increased, (ii) decreased, (iii) multiplied and, (iv) divided by the same quantity (except zero), the resulting series is also in A.P