303.有一系列数c, c^2,c^3…C^n 问the sum of these number is odd?

leeguoyan

303.有一系列数c, c^2,c^3…C^n  问the sum of these number is odd?
(1)    c is odd
(2)    n is odd
àC ?

liyang229

ok, at first, we know odd+odd=even, even+even=even, only and only if even +odd = odd.
right?
and also, we know, evenxeven=even, odd xeven=even, only oddxodd =odd.
so, this become interesting now.
for x^x, we could make is become x*x*x*x.......*x.  and the total number of x should be and odd number in order to make x^x odd. otherwise, it will be an even number.

Therefore, in the question, we need to identify if the sum of c, c^2,c^3…C^n  is odd.

ok, for statement 1. c is odd. so, c^2 will be odd, c^3 will also be odd. and c^n, will also be odd, because we said, odd*odd is always odd.
so, at this point, we will know all the terms in the list are odd numbers.
but, as what we talked about at the beginning,only and only if even +odd = odd. which means  there must  be a even number.  and also, we said, odd+odd =even. so, if the number of terms is even, the sum of these terms will become even.

so, the function of statement 2 is coming now, n is odd. that means the number is terms in the list is odd, which means it is even+1.

as what we said, when the terms number is even, the sum of odd number is even.

so, for the sum of n-1 term, it is even, right?

then, the last term, is also a odd number.

which then makes the final sum becomes  even+odd. then the final sum is an odd number.


ok, lets take a breath and finally say, we have enough evidence to prove the sum of  c, c^2,c^3…C^n  is odd.

stephani

Is it necessary to say that "c" is an integer?

liyang229

decimal number cannot be odd or even.

leeguoyan

thanks a lot

py_0000

Thanks a lot~