Clarification on this Question (1 Viewer)

[HSCya1234567]

Hi all, this might seem really simple, but I was just reading through the solution to this paper, I wasn't sure why k +1 is a factor of (2k + 1)! i understand 2k because of the 'unrolling' of the factorial

 

[HSCya1234567]

Question if needed

Define the following

P = 1 X 2 X 3 X ... n

S = 1 + 2 + 3 ... n

Show that if n is odd, then P is divisible by S.

I said that P was n! and S was n(n+1)/2 and then let n = 2k + 1 and then divided the two to get what they have in the answers

 

[SadCeliac]

Hi all, this might seem really simple, but I was just reading through the solution to this paper, I wasn't sure why k +1 is a factor of (2k + 1)! i understand 2k because of the 'unrolling' of the factorial

View attachment 39173

Maybe because as you unroll (2k + 1)! you end up getting (k + 1)? That's my guess:

(2k + 1)! = (2k + 1)(2k)(2k - 1)(2k - 2)...(k + 1)(k)(k - 1)...(3)(2)(1)

Maybe that's why?

 

[HSCya1234567]

Can the 2 just disappear though?

 

[SadCeliac]

Can the 2 just disappear though?

I mean... if you let k = 2, for example:

2k + 1 = 5, so we are unrolling 5!

Your (k + 1) in this case is 3, which exists in the following unrolling step...

I.e., (2*2 + 1)! = (2*2 + 1)(2*2)(2*2 - 1)(2*2 - 2)(2*2 - 3) = 5x4x3x2x1

Which is why I assume the 2 'disappears', because eventually you'll have subtracted enough from (2k + 1) to get a (k + 1)

 

[HSCya1234567]

oooh okay wait i think that makes sense thankyou so much!

 

[SadCeliac]

oooh okay wait i think that makes sense thankyou so much!

no worries!