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.

nth TERM OF AN A.P. : It is denoted by tn and is given by the formula, tn = 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 a1 , a2, , an, we have d= ak + 1 — ak where ak + 1 and ak are the (k+ 1)th and the kth terms respectively.

SUM OF FIRST N TERMS OF AN A.P.

It is represented by symbol Sn and is given by the formula,

Sn= n/2{ 2a + (n — 1)d} or, Sn = 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 — an = Sn— Sn – 1.

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

nth 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 Sn= n(n + 1) / 2

Click Here for All Maths Class 10 Notes

You wish to report grammatical or factual errors within our online articles, you can let us know using the article feedback form.

comments