Definition:
Let f:N->R be a function and let f(n)=an. Then, a1,a2,a3,...,an,.... is called the sequences in R deter mind by the function f and it is denoted by (an). an is called n-th term of the sequences.
RANGE:
The range of the function f, which is a subset of R is called the range of the sequence.
Examples:
i) The f:N->R given by f(n)=n-power 2. Determind the sequence 1,4,9,....n-power 2,....
ii) f(n)=n determind the sequence 1,2,3,.....,n,......
iii) f:N->R given by f(n)=(-1)power n -1,1,-1,1,.....(-1)power n,.........
The range of the sequence is {1,-1}.
The range of the sequence may be finite or infinite.
iv) (-1)power n+1 determind sequence, 1,(-1),1,(-1),....(-1)power n+1,......
v) The constant function f:N->R given by f(n)=1 determind sequence.
1,1,1,1,.........
Such a sequence is called constant sequence.
vi) f(n)={n/2 if n is even
{1/2 (1-n) if n is odd
sequence, 0,1,-1,2,....n,-n,....
vii) Let x belongs to R the f:N->R given by, f(n)=x power n-1 determind Sequence.
1,x,x power 2,...........x power n-1,....
is called Geometric Sequence.
viii) Let a1=1, a2=2 and an=an-1+an-2 determind the sequence.
1,1,2,3,5,8,13,21,34,............ is called fibonacci series.
Let a1= root 2 and an+1= root of (2+an) determind the sequence.
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