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

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