If a_n=2^n/(2n)! for intergers n > 0, then the ratio (a_(n+1))/a_n is equal to what?

Question

a) 2/(2n+1)

b)2/n

c)2^(n+1)/2(n+1)

d)2(n+1)/2

e)1/[(n+1)(2n+1)]

Answer 1

e.

Work it out by actually writing out the problem, and then simplify.

2^(n+1) * (2n)!

------------------------- =

2^(n)    * (2(n+1))!

2^(1) * 1*2*3...*2n

--------------------------------------...  =      

(1)    * 1*2*3...*2n*(2n+1)*(2n+2)

2

---------------------

(2n+1)*2*(n+1)

1

----------------

(2n+1)(n+1)