## Maths Class 10 Notes for Arithmetic Progressions

**SEQUENCE**

A collection of numbers arranged in a definite order according to some definite rule (rules) is called a sequence.

Each number of the sequence is called a term of the sequence. The sequence is called finite or infinite according as the number of terms in it is finite or

infinite.

**ARITHMETIC PROGRESSION**

A sequence is called an arithmetic progression (abbreviated A.P.) if and only if the difference of any term from its preceding term is constant.

A sequence in which the common difference between successors and predecessors will be constant. i.e. a, a+d,a+2d

This constant is usually denoted by ‘d’ and is called common difference.

**NOTE :** The common difference ‘d’ can be positive, negative or zero.

**SOME MORE EXAMPLES OF A PARE**

(a) The heights (in cm) of some students of a school standing in a queue in the morning assembly are 147, 148, 149, ….. , 157.

(b) The minimum temperatures (in degree celsius) recorded for a week in the month of January in a city, arranged in ascending order are 3. 1, — 3. 0, — 2. 9, — 2. 8, — 2.7, — 2. 6, — 2. 5

(c) The balance money (in ) after paying 5% of the total loan of Z 1000 every month is 950, 900, 850, 800, ….50.

(d) The cash prizes (in ₹) given by a school to the toppers of Classes Ito XII are, respectively, 200, 250, 300, 350„ 750.

(e) The total savings (in ₹) after every month for 10 months when Z 50 are saved each month are 50, 100, 150, 200, 250, 300, 350, 400, 450, 500.

**n ^{th} TERM OF AN A.P. :** It is denoted by t

_{n}and is given by the formula, t

_{n}= a + (n —1)d

where ‘a’ is first term of the series, n is the number of terms of the series and ‘d’ is the common difference of the series.

**NOTE :** An A.P which consists only finite number of terms is called a finite A.P. and which contains infinite number of terms is called infinite A.P.

**REMARK :** Each finite A.P has a last term and infinite A.Ps do not have a last term.

**RESULT:** In general, for an A.P a_{1} , a_{2}, , a_{n}, we have d= a_{k + 1} — a_{k} where a_{k + 1} and a_{k} are the (k+ 1)th and the kth terms respectively.

**SUM OF FIRST N TERMS OF AN A.P.**

It is represented by symbol S_{n} and is given by the formula,

S_{n}= n/2{ 2a + (n — 1)d} or, S_{n} = n/2 { a + l} ; where ‘l’ denotes last term of the series and l= a+(n-1)d

**REMARK :** The nth term of an A.P is the difference of the sum to first n terms and the sum to first (n — 1) terms of it. — ie — a_{n} = S_{n}— S_{n – 1}.

**TO FIND nth TERM FROM END OF AN A.P. : **

n^{th} term from end is given by formula l – (n – 1)d

nth term from end of an A.P. = nth term of (l, l — d, l – 2d,…….)

=l+(n-1)(—d)=l—(n-1)d.

**PROPERTY OF AN A.P. :**

If ‘a’ , b, c are in A.P., then

b — a= c — b or 2b= a + c

**THREE TERMS IN A.P. : **

Three terms of an A. P. if their sum and product is given, then consider

a—d,a,a+d.

**FOUR TERMS IN A.P. : **

Consider a —3d, a — d, a+ d, a +3d.

**NOTE :**

The sum of first n positive integers is given by S_{n}= n(n + 1) / 2