[SOLVED] induction integer question

**Khonics89** · October 7th 2009, 08:26 PM

Prove that for any integer n ≥ 1,
(2n)!/2^n is an integer!

Well I started off saying let n=(2n!)/2^n (*)

and did the usual step like let n=1 (which is obviously true)

then when I go to let n=n+1

i get

n + (n+1) = (2(n+1))!/2^(n+1)

i can substitute the LHS n with (*)

but then I get stuck and dont know what to do

so im guessing my method is incorrect :\

can someone shed some light on this.. thanks!

**Soroban** · October 7th 2009, 09:46 PM

Hello, Khonics89!

Prove that for any integer $n \geq 1,\;\frac{(2n)!}{2^n}$ is an integer.

You started correctly . . .

$S(n)\!:\;\;\frac{(2n)!}{2^n} \:=\:a,\:\text{ for some integer }a.$

Verify $S(1)\!:\;\;\frac{2!}{2^1} \:=\:1$ . . . True.

Assume $S(k)$ is true: . $\frac{(2k)!}{2^k} \:=\:a,\:\text{ for some integer }a.$

We want to prove that $S(k+1)$ is true: . $\frac{(2[k+1])!}{2^{k+1}}$ is an integer.

We have: . $\frac{(2[k+1])!}{2^{k+1}} \;\;=\;\;\frac{(2k+2)!}{2^{k+1}} \;=\;\frac{(2k+2)(2k+1)(2k)!}{2\cdot2^k}$

. . $= \;\;\frac{(2k+2)(2k+1)}{2}\cdot\frac{(2k)!}{2^k} \;\;=\;\;\frac{2(k+1)(2k+1)}{2}\cdot\frac{(2k)!}{2 ^k}$

. . $=\;\;(k+1)(2k+1)\cdot\underbrace{\frac{(2k)!}{2^k} }_{\text{This is }a} \;\;=\;\;\underbrace{(k+1)(2k+1)\!\cdot\!a }_{\text{This is an integer}}$

We have proved $S(k+1)$ . . . The inductive proof is complete.

**Khonics89** · October 8th 2009, 05:03 PM

Thanks- yeah I never saw it like that

basicaly I did the proof but didn't know it was right .. soo yeah thanks once again

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