# Arithmetic Progression - Class 10 : Notes

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

**Â **

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

**1, 8, 27, 64, 125,â€¦â€¦**

*For Example: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.
**

**2, 4, 6, 8,â€¦. is a arithmetic progression because number are even natural numbers where**

*For Example:Â***(3) If â€˜aâ€™ is the first term and 'd' the common difference of an AP,Â then the A.P.Â isÂ
**

**If AP is 2, 4, 6, 8,â€¦. Then first term and**

*For Example:Â*So, .

**(4) A sequence is an AP, if is independent of n.
**

**If sequence is 2, 4, 6, 8, â€¦â€¦ ,â€¦.. so if we take so Â So which is independent of n.**

*For Example:Â***(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.
**

**A sequence 1, 4, 9, 16, 25,â€¦. Is an AP. Suppose term which is a linear expression in n. which is .**

*For Example:Â***(6) The term , of an AP with first term â€˜aâ€™ and common difference â€˜dâ€™ is given byÂ
**

**If want to find Â term in example given in 4**

*For Example:Â*^{th}.

, then we can find 10

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

^{th}term ofÂ sequence is

**(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.**

**Determine the 10**

*For Example:Â*^{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.

**(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.
**

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

*For Example:Â***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)

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 the A.P**

*For Example:*

Now

and also

We have

Or

Hence common difference =

Awesome

Very very very usefu

Plese me sir math