Bharat Gupta is teaching live on Unacademy Plus
SEQUENCE AND SERIES Course Overview
Sequence sbjects that follows a certain pattfrnu.mbers or An ordered set of objects or numbers, like a , , , a4, a5, a6 per certain rule, has a definite value. an are said to be in a sequence, if, as Series series refers to the sum of the elements of the sequence The addition of the terms of a sequence (an), is known as series. Like sequence, series can also be finite or infinite, where a finite series is one that has a finite number of terms written as al + a2 + a3 + as + as a+..a Finite Sequence/Series: A finite sequence is one that stops at the end of the list of numbers al, a2, , a4, a5, a6 an. Infinite Sequence/Series: An infinite sequence refers to a sequence which is unending, a1, a2, , a4, a5, a6
Key Differences Between Sequence and Series The sequence is defined as the collection of numbers or objects that follow a definite pattern. When the elements of the sequence are added together, they are known as series. Order matters in a sequence, as there is a certain rule that prescribes the pattern of the sequence. Hence, 2, 5, 8 is different from 5, 2, 8. On the other hand, in a series order of appearance may or may not matter, like in the case of absolutely convergent series the order doesn't matter. So, 258 is same as 5 +2 8, only their sequence is different.
Arithmetic Progression (A.P.) and Geometric Progression (G.P) . . and G.P. are also sequences, not series. Arithmetic Progression is a sequence in which there is a common difference between the consecutive terms such as 1, 3, 5, 7 and so on. On the contrary, in a geometric progression, each element of the sequence is the common m ultiple of the preceding term such as 2, 4, 8, 16 and so on. Sim ilarly, Fibonacci Sequence is also one of the popular infinite sequence, in which each term is obtained by adding up the two preceding term s 1, 2, 3, 5, 8, 13, 21 and so on
SOME EXAMPLES OF SEQUENCE (i) 1, 4,7, 10,. 19 Here each term is obtained by adding 3 to the previous term (ii) 2, -4, 8, - 16, Here each term is obtained by multiplying the preceding term by - 2 (iii) 2, 3 5, 7, 11, 13 This is the sequence of prime numbers. Note that sequence i Safin le sequence whereas others are infinite sequences. PROGRESSIONS If the terms of a sequence follow certain pattern, then the sequence is called a progression. Here we shall study three special types of progressions: (i) Arithmetic Progression (A. P.) (ii) Geometric Progression (G.P.) (iii) Harmonic Progression (H.P.)
In this course you wil learn nth term, Sum of n terms, sum of infinite terms of (i) Arithmetic Progression (A. P.) (ii) Geometric Progression (G.P.) ii) Harmonic Progression (H.P.) (iii) Harmonic Progression (H.P.) (iv) Special Progression