Question

Is 2k-1 odd?
I get that 2(some int k) + 1 is the property for odd numbers.
The main question:
I am confused on how 2k-1= 2k-2+1 which is a form of k?

Answer 1

About , the simple logic is that, if one want to express even numbers, the formula for the series is , and the table would be as below.

k	1	2	3	. . .
2k	2	4	6	. . .

What is important is, that k specifies the position of the series, as the kth even integer is , like 1st is 2, second is 4, third is 6, and so on. Also, as the odd numbers are adjacent to the even numbers, the likely formula for odd numbers would be either or . Both are correct, but one is a bit more correct than the other. As we would like to express the formula as the kth odd number, the 1st one would be 1 if is considered, while 1st one is supposed to be 3 if is considered. What can be done is that, the kth odd integer is taken as , and (k+1)th odd integer is taken as . The further series can be verified indeed, as in the table below.

k	1	2	3	. . .
2k-1	1	3	5	. . .
k	0	1	2	. . .
k+1	1	2	3	. . .
2k+1	1	3	5	. . .

In considering , k should be started from 0, as the formula expresses (k+1)th odd integer.

By means of formul, we can prove that as for or or or or . The thing is, as , obviously expresses kth odd integer.

Also, we can logically infer as, taking as the expression of odd numbers, the expression would be to express the (k+1)th odd integer, and the just previous to (k+1)th term is (k+1 - 1)th term or the kth term. In that sense, the expression would express the kth term or (k+1 - 1)th term as or or . The thing is, to access the odd numbers via positions, we have to go through the even terms which are expressed by 2k. We can either reach the previous term of 2k (which is ) or the next term of 2k (which is ). Whatever we choose, we must see by arguiing whether the expression expresses kth term or the (k+1)th term.