# Arithmetic Progression - Class 10 : Notes

(1) A sequence is an arrangement of numbers or objects in a definite order.*For Example: *1, 8, 27, 64, 125,……

Above arrangement numbers are arranged in a definite order according to some rule.

(2) A sequence *For Example: *2, 4, 6, 8,…. is a arithmetic progression because number are even natural numbers where

(3) If ‘a’ is the first term and 'd' the common difference of an AP, then the A.P. is *For Example: *If AP is 2, 4, 6, 8,…. Then first term

So,

(4) A sequence *For Example: *If sequence is 2, 4, 6, 8, ……

(5) A sequence *For Example: *A sequence 1, 4, 9, 16, 25,…. Is an AP. Suppose

(6) The *For Example: *If want to find

(7) Let there be an A.P with first term ‘a’ and common difference d. if there are m terms in the AP, then

term from the beginning =

Also,

= *For Example: *Determine the 10th term from the end of the A.P 4, 9, 14, …, 254.

(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 ‘a’ and common difference ‘d’ is given by *For Example: *(i) 50, 46, 42, … find the sum of first 10th term

Solution:

Given,

Here , first term ,

Difference

And no of terms

We know

Hence, Sum of 10 terms is 320.

(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 AP’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: *The ratio of the sum of n terms of two AP’s is (7n+1):(4n+27). Find the ration of their terms.

Solution:

let , be the 1st terms and , the common differences of the two given A.P’s. then the sums of their n terms are given by,

and

It is given that

...........(i)

To find ratio of the terms of the two given AP’s, we replace by in equation (i). Therefore,

Hence, the ratio of the terms of the two AP’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, B are constants. In such a case the common difference is 2A.*For Example:*For the A.P

Now

and also

We have

Or

Hence common difference =

Notes for arithmetic progression chapter of class 10 Mathematics. Dronstudy provides free comprehensive chapterwise class 10 Mathematics notes with proper images & diagram.

**(1) A sequence is an arrangement of numbers or objects in a definite order.**

* For Example: *1, 8, 27, 64, 125,……

Above arrangement numbers are arranged in a definite order according to some rule.

**(2) A sequence a1,a2,a3,....,an,.. is called an arithmetic progression, if there exists a constant d such that, a2−a1=d,a3−a2=d,a4−a3=d,...,an+1−an=d and so on. The constant d is called the common difference.**

* For Example: *2, 4, 6, 8,…. is a arithmetic progression because number are even natural numbers where a1=2,a2=4,a3=6,a4=8

4−2=2,6−4=2,8−6=d,...,an+1−an=2

**(3) If ‘a’ is the first term and 'd' the common difference of an AP, then the A.P. is a,a+d,a+2d,a+3d,a+4d....**

* For Example: *If AP is 2, 4, 6, 8,…. Then first term a=2 and d=2

So, 2,2+4,2+2(2),2+3(2),a+4(2).....

**(4) A sequence a1,a2,a3,....,an,.. is an AP, if an+1−an is independent of n.**

* For Example: *If sequence is 2, 4, 6, 8, …… an,….. so if we take an=16 so an+1=18 So an+1−an=18−16=2which is independent of n.

**(5) A sequence a1,a2,a3,....,an,.. is an AP, if and only if its nth term an is a linear expression in n and In such a case the coefficient of n is the common difference.**

* For Example: *A sequence 1, 4, 9, 16, 25,…. Is an AP. Suppose nth term an=81 which is a linear expression in n. which is n2.

**(6) The nth term an, of an AP with first term ‘a’ and common difference ‘d’ is given by an=a+(n−1)d**

* For Example: *If want to find nth term an in example given in 4

^{th}.

a=2, d=2 then we can find 10

^{th}term by putting n=10 in above equation. So 10

^{th}term of sequence is a10=2+(10−1)2=20

**(7) Let there be an A.P with first term ‘a’ and common difference d. if there are m terms in the AP, then**

** nth term from the end = (m−n+1)th**

**term from the beginning =a+(m−n)d**

**Also, nth term from the end = Last term + (n−1)(−d)**

**= l−(n−1)d, where l denotes the last term.**

* For Example: *Determine the 10

^{th}term from the end of the A.P 4, 9, 14, …, 254.

l=254, d=5

nth term from the end =l−(10−1)d = l−9d= 254−9×5=209

**(8) Various terms is an AP can be chosen in the following manner.**

Number of termsTermsCommon difference3a−d,a,a+dd4a−3d,a−d,a+d,a+3d2d5a−2d,a−d,a,a+d,a+2dd6a−5d,a−3d,a−d,a+d,a+3d,a+5d2d

**(9) The sum to n terms of an A.P with first term ‘a’ and common difference ‘d’ is given by Sn=n2{2a+(n−1)d} Also, Sn=n2{a+l}, where l= last term = a+(n−1)d**

**For Example: ****(i) 50, 46, 42, … find the sum of first 10**^{th}** term**

**Solution:**

Given, 50,46,42,.....

Here , first term ,

Difference

And no of terms

We know

Hence, Sum of 10 terms is 320.

**(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 AP’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: *The ratio of the sum of n terms of two AP’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.P’s. then the sums of their n terms are given by,

and

It is given that

...........(i)

To find ratio of the terms of the two given AP’s, we replace by in equation (i). Therefore,

Hence, the ratio of the terms of the two AP’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, B are constants. 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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