Question

How would I show the following or prove:

Answer 1

isnt that a musical symbol the right-most symbol? :P

Answer 2

You mean the treble cleff?

Answer 3

o.O ??

Answer 4

In your equation:\[\prod_{r=1}^n(2r-1)=\frac{n!}{2^n}\left(\frac{2n}{n}\right)\]what does the term:\[\left(\frac{2n}{n}\right)\]imply?

Answer 5

does it mean:\[\frac{n!}{2^n}\left(\frac{2n}{n}\right)=\frac{n!}{2^n}\times\frac{2n}{n}=\frac{n!}{2^n}\times\frac{2}{1}=\frac{n!}{2^{n-1}}\]

Answer 6

@asnaseer that means combination(2n,n) or C(2n,n) or (2n)! /( n!)^2 etc

Answer 7

thx @shubhamsrg , so this means we are trying to prove:\[\prod_{r=1}^n(2r-1)=\frac{n!}{2^n}\left(\frac{2n}{n}\right)=\frac{n!}{2^n}\times\frac{(2n)!}{n!(2n-n)!}=\frac{\cancel{n!}}{2^n}\times\frac{(2n)!}{\cancel{n!}n!}=\frac{(2n)!}{2^nn!}\]

Answer 8

hmmn..

Answer 9

I wonder if we can use induction here?

Answer 10

are we allowed to use induction even if the question doesnt say so?

Answer 11

why not?

Answer 12

if the ques doesnt mention anything,,i guess we should be free to use any mathematical tool..and induction is one such..but maybe it might get complicated a bit..well lets try,,

Answer 13

if we use induction, we are just proving that the equation is true, how would i show that the LHS =RHS?

Answer 14

thats the point of induction - it proves the assertion stated by the equation is true

Answer 15

ok, i ll go with that then. Thanks

Answer 16

yw

Answer 17

I've managed to prove it by induction - so let me know if you get stuck and I'll give some pointers.