Question

Prove that the sequence 2ncn is increasing

Answer 1

Answer:

2n!)

​  

=2  

n

(1.3.5...(2n−1))

RHS =(1.3.5...(2n−1))2  

n

 

Multiply and divide by 2.4.6...2n on RHS

[1.3.5...(2n−1)]2  

n

=(1.3.5...(2n−1))2  

n

×  

2.4.6....2n

2.4.6...2n

​  

 

=  

2.4.6....2n

[1.3.5...(2n−1)]2  

n

 

​  

 

=  

2.4.6....2n

[(2n)(2n−1)....5.4.3.2.1]2  

n

 

​  

 

=  

2.4.6....2n

(2n)!2  

n

 

​  

                           

=  

2  

n

(1.2.3..n)

(2n)!2  

n

=  

n!

(2n)!

​  

= LHS .proved