Ch 6 Flashcards
Sequence definition
Sequence is a function whose domain is a sub set of the set of natural numbers
Real sequence definition
If all members of a sequence or real numbers then it is called a real sequence
Finite and infinite sequence
If the domain of a sequence is a finite set then the sequence is called finite sequence otherwise and infinite sequence. An infinite sequence has no last term.
Progression definition
If the times of a sequence follow a certain pattern then the sequence is called a progression
Arithmetic progression (A.P)
A sequence is called arithmetic sequence or arithmetic progression if the difference of any two consecutive terms is the same.
Common difference
Let a and b and AP then the common difference is donated by d and is defined as d=a(n) - A(n-1)
Arithmetic mean (A.M)
A number “A” is said to be the arithmetic mean between two numbers A and b if a, A, b are in arithmetic progression.
I.e A= a+b/2
Series
The sum of an indicated number of times in a sequence is called a series let A1,A2,A3 be a sequence then A1 + A2 + A3+….. is called a series
Geometric progression (G.P)
A sequence {an} is called geometric sequence or geometric progression if ratio of any two consecutive terms of {an} is the same.
Common ratio
Let be a geometric progression then combination is denoted by r and is defined as r= an/an-1
Geometric mean
A number G is said to be geometric mean between two numbers A and b if a, G, b are in genetic progression.
I.e G.M= ±√ab
Convergent series
If Sn –> l as n–> infinity, then series is said to be convergent
Divergent series
If Sn increases or decreases indefinitely as n –> infinity then we say that Sn does not exist and the series is said to be divergent
Harmonic progression (H.P)
A sequence of numbers is called a harmonic progression if the reciprocals of its terms are in A.P.
If 1/2, 1/3 are in H.P then 2,3 are in A.P
Harmonic mean ( H)
A number H is said to be the harmonic mean between two numbers a and b if a, H and b are in H.P
I.e H= 2ab/a+b
Relation between arithmetic, geometric and harmonic means
A×H= G²
Arithmetic sequence find terms formula
an=a1+(n-1)d
1+2+3+…..n =
n(n+1)/2
Sum of arithmetic series
When you know first and last term
Sn=n/2{a1+an}
When you know a1,n,d or sum
Sn=n/2{2a1+(n-1)d}
Geometric progression formula
an=a1 r^n-1
How to break a4 in geometric progression
a4=a1r^3
Ingematic progression common ratio formula
r=a2/a1
Sum of infinite geometric progression
S∞= a1/1-r
Sum of finite geometric progression
Sn= { a1(1-r^n)}/1-r
