Notes for arithmetic progression chapter of class 10 Mathematics. Dronstudy provides free comprehensive chapterwise class 10 Mathematics notes with proper images & diagram.
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(1) A sequence is an arrangement of numbers or objects in a definite order.
For Example:Â 1, 8, 27, 64, 125,â€¦â€¦
Above arrangement numbers are arranged in a definite order according to some rule.
(2) A sequence is called an arithmetic progression, if there exists a constant d such that,Â and so on.Â The constant d is called the common difference.
For Example:Â 2, 4, 6, 8,â€¦. is a arithmetic progression because number are even natural numbers where
(3) If â€˜aâ€™ is the first term and 'd' the common difference of an AP,Â then the A.P.Â isÂ
For Example:Â If AP is 2, 4, 6, 8,â€¦. Then first term and
So, .
(4) A sequence is an AP, if is independent of n.
For Example:Â If sequence is 2, 4, 6, 8, â€¦â€¦ ,â€¦.. so if we take so Â So which is independent of n.
(5) A sequence is an AP, if and only if its term is a linear expression in n and In such a case the coefficient of n is the common difference.
For Example:Â A sequence 1, 4, 9, 16, 25,â€¦. Is an AP. Suppose term which is a linear expression in n. which is .
(6) The term , of an AP with first term â€˜aâ€™ and common difference â€˜dâ€™ is given byÂ
For Example:Â If want to find Â term in example given in 4^{th} .
, then we can find 10^{th} term by putting n=10 in above equation. So 10^{th} term ofÂ sequence is
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(7) Let there be an A.P with first term â€˜aâ€™ and common difference d. if there are m terms in the AP, then
Â term from the end =
term from the beginningÂ =
Also,Â term from the end = Last term +
= , whereÂ denotes the last term.
For Example:Â Determine the 10^{th} term from the end of the A.P 4, 9, 14, â€¦, 254.
,
term from the end = = = =209
(8) Various terms is an AP can be chosen in the following manner.
Number of terms | Terms | Common difference |
3 | d | |
4 | 2d | |
5 | d | |
6 | 2d |
(9) The sum to n terms of an A.P with first term â€˜aâ€™ and common difference â€˜dâ€™ is given byÂ Â Also, , where = last term =
For Example:Â (i) 50, 46, 42, â€¦ find the sum of first 10^{th} term
Solution:
Given,
Here ,Â first term ,
Difference
And no of terms
We know
Hence, Sum of 10 terms is 320.
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(ii) First term is 17 and last term is 350 and d=9 so find total sum and find how many terms are there.
Solution:
Given, first term, a=17, last term, = 350 =
And difference d = 9
We know,
We know, sum of n terms
Hence, number of terms is 38 and sum is 6973.
(10) If the ratio of the sums of n terms of two APâ€™s is given, then to find the ratio of their terms, we replace n by (2n-1) in the ratio of the sums of n terms.
For Example:Â The ratio of the sum of n terms of two APâ€™s is (7n+1):(4n+27). Find the ration of their terms.
Solution:
let , be the 1^{st} terms and , the common differences of the two given A.Pâ€™s. then the sums of their n terms are given by,
and
It is given that
...........(i)
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To find ratio of the terms of the two given APâ€™s, we replace by in equation (i). Therefore,
Hence, the ratio of the terms of the two APâ€™s is
So as per rule if we replace by we get ratio
(11) A sequence is an AP if and only if the sum of its n terms is of the form , where A,Â B are constants.Â In such a case the common difference is 2A.
For Example:
For the A.P
Now
and also
We have
Or
Hence common difference =
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