SEQUENCE AND SERIESBY ABHISHEK SAXENA : SEQUENCE AND SERIESBY ABHISHEK SAXENA FOR THE STUDENT WHO ARE APPEARING FOR ENGINEERINGENTRANCE EXAMS
WHAT IS A SEQUENCE ? : WHAT IS A SEQUENCE ? A function whose domain is the set of natural numbers is a sequence.
We can also say that a function with domain N range R is called a real sequence.
Ex.:
1,2,3,4,5,…….
2,4,6,8,10,…….
1,3,5,7,9,………
1,4,9,16,25,…….
WHAT IS A SERIES ? : WHAT IS A SERIES ? If a1,a2,a3,a4,…..,an is a sequence then
a1+a2+a3+a4+….. +an will be a series.
Ex. 1,3,5,7,9,11,13,…….. is a sequence and
1+3+5+7+9+11+13+…….. is a series.
WHAT IS A PROGRESSION ? : WHAT IS A PROGRESSION ? The progression is a sequence whose terms follow certain patterns.
Types of progression:
ARITHEMETIC PROGRESSION (A.P.)
GEOMETRIC PROGRESSION (G.P.)
ARITHEMETIC PROGRESSION (A.P.) : ARITHEMETIC PROGRESSION (A.P.) a,a+d,a+2d,a+3d,a+4d,…………+a+(n-1)d
is an (A.P.) of n terms.
1. General term = a+(n-1)d
nth term from the end of a arithmetic progression having m terms : a+(m-n)d
Sum of n terms of a A.P.
Sn= (n/2)(2a+(n-1)d)
= (n/2)(a+l) Where l is the last term of that A.P.
GEOMETRIC PROGRESSION (G.P.) : GEOMETRIC PROGRESSION (G.P.) a,ar,ar2,ar3,ar4,…..,ar(n-1) is a G.P. of n terms.
nth term of a G.P.= ar(n-1)
nth term from the end of a G.P. having m terms= ar(m-n)
Sum of n terms of a G.P.=
Sn=a(rn-1)/(r-1) when r>1 and r is not equal to1
=a(1-rn)/(1-r) when r<1 and r is not equal to1
Sum of an infinite G.P.=a/(1-r)
GEOMETRIC PROGRESSION (G.P.) : GEOMETRIC PROGRESSION (G.P.) If a1+a2+a3+a4+……. +an is a arithmetic progression then 1/a1+1/a2+1/a3+1/a4+…..+1/ an is a Harmonic progression.
Ex.
2+4+6+8+10+12+…………. is a arithmetic progression then
½+1/4+1/6+1/8+1/10+1/12+…………is a Harmonic progression.
Relation between A.M.,G.M.,H.M. : Relation between A.M.,G.M.,H.M. A.M.>G.M.>H.M.
(G.M.)2=(A.M.)*(H.M.)