Double factorial formula

Question

I don't know why this is true:

$$(2k+1)!!=\frac{(2k+1)!}{2^kk!}$$

Can anyone explain it for me? I know what is double factorial, but would like to know how this formula was derived. Thanks.

Answer 1

By multiplying both sides by $(2k)!! = (2k)(2k-2)\cdot\ldots\cdot 2 = 2^k\cdot k!$, you get: $$(2k+1)! = (2k+1)!.$$

Answer 2

HINT : $$(2k+1)!!=1\cdot 3\cdot 5\cdot \cdots (2k-1)\cdot (2k+1)$$ $$(2k+1)!=1\cdot 2\cdot 3\cdot \cdots (2k)\cdot (2k+1) $$ $$2^k\cdot k!=2^k\times \{1\cdot 2\cdot 3\cdot \cdots (k-1)\cdot k\}=2\cdot 4\cdot 6\cdot\cdots (2k-2)\cdot (2k).$$