Do you mean that (n+1)(n+2)...(2n) is divisible by 4? This is obvious because 2n is even and, since the number of factors is ≥ 3 for n ≥ 3, there is another even factor.
Hi,
I could use some help with my proving. The assignment goes as follows:
"Prove with mathematical induction that
((n+1)(n+2)...(2n)) / 2^{2}
is a whole number with all values of n"
I know the basic principle of induction but I can't see how can I show the "n+1" -step.
Please excuse my inaccurate expressions, my native language is finnish I would REALLY appreciate the help!
Do you mean that (n+1)(n+2)...(2n) is divisible by 4? This is obvious because 2n is even and, since the number of factors is ≥ 3 for n ≥ 3, there is another even factor.
What, exactly, do you mean by ((n+1)(n+ 2)...(2n)) if n= 1? I see that, for example, if n= 3, then this is 4(5)(6) but what about n= 1? Is this "2" or "2(2)"? If the first, which is what I would think because 1+ 1= 2(1), the statement is not true:
But for n= 2, this is 3(4)(4) which is divisible by because it has a factor of 4. If , there are four consecutive numbers in that so at least one is divisible by 4.
I chose to write the induction hypothesis after looking at the first several statements:
We easily see that is true, so next I defined:
Now, adding to both sides of there results:
We have derived from thereby completing the proof by induction.