Sequence: A set of numbers arranged in some definite order and formed according to some rule is called a sequence.
The different numbers in a sequence are called its terms which are generally denoted by a1, a2, a3, a4, a5…… an etc. which means, 1st term, 2nd term …. nth term. The nth term is also called General Term of the sequence.
A sequence can be represented in many ways, e.g. 2, 4, 6, 8, ……. Whose nth term is 2n for all natural number ‘n’.
A sequence can also be represented by writing the nth term straight way as an = 2n + 5, from here, we can find the various terms of the sequence by giving values to n from 1, 2, 3, …. . So the resulting sequence is 7, 9, 11, ….
Series: If a1, a2, a3, a4, a5…… is a sequence then a1 + a2 + a3 + a4 + a5…… is a series.
Progression: The sequence that follows a certain pattern is called a progression.
Arithmetic Progression: It is a sequence in which the successive terms increase or decrease by a common difference is known as arithmetic progression and is written as A.P
Examples:
1) 1, 5, 9, 14, 17, ……
2) 1, 2, 3, 4, 5, …..
Common difference of an AP: The difference between any successive members is a constant and it is called the common difference of AP
1) If a1, a2, a3, a4, a5 are the terms in AP, then
d = a2 – a1, a3 – a2, a4 – a3, a5 – a4
2) We can represent the general form of AP in the form
a, a + d, a + 2d, a + 3d, a + 4d, …….
where ‘a’ is first term and ‘d’ is the common difference
3) nth term of Arithmetic Progression:
nth term = a + (n – 1) d
4) Sum of the nth term in Arithmetic Progression:
Sn = n/2[a + (n – 1) d]
Sn = n/2 [t1 + tn]
Sn = n/2 [a + l] , where a is first term and l is the last term
5) If a, b, c are in AP, then
b = (a + c)/2