shrouded1
vicksikand
If its a sequence be it an Arithmetic, Geometric or Harmonic progression - it has to have some order.
Given S is an infinite Sequence of positive Integers.
1: 1st 10 in S are even - makes you wonder whether the sequence has infinite even's.
2,4,6,8,10,12....2k
or it could be 4,10,16,...... a+6
I really couldn't find a way to introduce Odd numbers after the 1st 10 Evens. Unless there are special cases such as:
f=2k for 1<k<11 and
f=2k+1 for k>11. But then this wouldnt be a sequence , right?
2: Infin number of integers in S are odd : Not sufficient.
Any views?
A sequence of integers doesnt need to be an AP, GP, or HP necessarily. All you need is a well defined set of rules, in order to get a sequence. For instance the following is an example :
Sequence S, such that the ith element is given by :
\(s_i = 2i\) for i=1 to 10
\(s_i=4i+3\) for i>10
So this is an infinite sequence in which only the first 10 numbers are even and the rest are an infinite number of odd numbers
\(s_i = 2i\) for i=1 to 10
\(s_i=4i+3\) for i>10
This sounds more like a function than a sequence and the values of i for which the function is valid - formulate the domain.
Correct me if I am wrong:
In mathematical terms a progression and a sequence are one and the same.
The only series/progressions I am aware of are : AP,GP,HP,Fibonacci, Cauchy, and Farey.